Energy

Put focus into roleplaying, even if you're watching someone else do it. Focus yourself on what they are saying and doing, even if it's kinda boring, and project your body language and voice while you're acting in-character. Be much less high-intensity when simply describing your bonuses while rolling, or asking someone to pass the chips. This will create natural focus on the roleplaying aspect of the game.

Speak In-Character

Wherever possible, assume what others are doing is a prompt for in-character roleplaying. DM asks what you do? Turn to someone, and say 'Philius, methinks we should cross that bridge and by Everam and St George, charge those there gnolls with swords in hand. Once we have them subdued, we'll take some answers from them, so we must leave at least some alive! What say you, well-met friend?"

Even if people don't respond in-character, and instead shift it back ooc, roleplaying just happened. Keep doing it and others will soon follow.

Ham It Up

Your character is 'quiet' and 'reasonable' and NOPE. Your character is the hammiest of ham. He's a loud cliche. He is instantly identifiable as the tropes that make him up - and he defines the setting by his very presence. It's unfortunate - but humans love ham. They love it.. a lot. Be something simple and understandable and loud, and they will get with the program really fast.

You can do this by being a masterful actor and roleplayer with any character, even a non-hammy one, but it is easiest with ham, so ham I will advise. Your paladin isn't just a paladin who likes cheese and moonlit walks on the beach - he's SIR GALAHAD THE MIGHTY, SUBDUER OF THE PEASANTS, DEFENDER OF THE WOMENFOLK, AND HIS MOUSTACHE BRISTLES AT THE SLIGHTEST SUGGESTION OF DRAGONS.

'Big' traits tend to focus things on the roleplaying a lot faster. Simpler is easier for the audience to understand.

Find Allies

Find people who will respond to your dramatic offers. When you address people, address them first, so they respond in-character, and then immediately pull other people in. People ignore offers initially, but if something is already rolling, they'll get rolled in with it. Some people will instinctively resist roleplaying offers, for all kinds of reasons - learn to identify them too, and offer to them last, once the roleplaying scene has the most momentum.

Be good at plot

Being able to identify where the adventure is going will let you advance the plot during a roleplaying scene - which both speeds up the adventure and means the time spent on roleplaying won't cause a weak GM to not let you hit the end of it.

Roleplay during combat

'LOOK OUT, FARAMIR! THE GNOLL IS AT YOUR BACK!' 'Galahad charges at the gnoll attacking' moving mini 'faramir, and' rolls dice 'swings at it with his mighty sword.' By including both speech and roleplaying-description in amongst your mechanical actions, you partially negate the disconnect that happens during the mechanics-rich combat portions of sessions. Have to know what you are doing on your turn before your turn rolls around, or anti-roleplayers will complain your roleplaying is slowing things up if you are not clearly doing it faster than anyone else.

Additionally, being good at combat, and giving tactical advice in-character that leads to defeating enemies quickly, will give more time overall for non-combat-constrained roleplaying itself.

Occasionally, roleplay during others' turns - have Galahad shout an encouraging phrase at an opportune moment. This has to be rare, and well-timed, though - an advanced technique.

Be Heroic, or Dastardly

Again, ham. By being heroic, and roleplaying it hard, you make other people who are not roleplaying feel heroic. By being dastardly, and roleplaying it hard, you make other people feel heroic also who are not roleplaying. You're giving them some of your roleplaying energy in a way that feels good for them. Morally grey is, again, a tougher sell. Note this isn't 'good' or 'evil', it's more saturday morning cartoon than that. Snidely Whiplashi, or Dudley DoRight.

Incorporate the GM

Don't just roleplay at fellow players. Roleplay at NPCs. Treat them with importance, and give the GM offers to roleplay right back at you. All of this applies to the GM, too. Getting the GM on-board with roleplaying, especially if you can advance the story while doing so, will be a tremendous boon to your cause.

You're right, by the way. Premade adventures, split up groups, schedules, public venues, this stuff just kills roleplaying and really makes it quite hard - I literally could not design a better system to do so.

But even in those kind of circumstances, I have personally sparked roleplaying in some extremely tough crowds. You won't see a huge improvement - but even the tiniest bit of roleplaying can be a huge welcome to you if you're in a roleplaying drought, and if you play with regularly the same pool of people, you'll find people gravitating to you that appreciate roleplaying, perhaps even to the extent that people will fight to have you in their groups.

Overall, though, the roleplaying will be in many ways a simpler thing than the rare high level roleplaying you can get in a home group.

But it's certainly not impossible.

Just have the courage to keep trying and don't give up.

I don't know your player, but I know something about playing D&D in AL with ADHD. That's my hobby. I also know something about working with young adults with ADHD in structured settings. That's my profession.

We shouldn't be trying to diagnose your player over the internet, so let's stick with the facts as you've laid them out: a player is disruptive, attributes much/most/all of that to ADHD, and you want to try to help them play without being disruptive. None of that is up for debate.

First thing to think about are the specific strengths and weaknesses this brings to your player. (In my experience they're one and the same.) The flip-side of inattention is the ability to quickly task-switch. The flip-side of fidgeting is working tirelessly at a trivial mechanical task. The flip-side of outbursts is spontaneous creativity.

With those in mind it's time to think about how an AL D&D game works, how it interfaces with those, how (much) to highlight areas where D&D plays to a strength, and how to mitigate those situations where D&D plays to a weakness.

• Some games feature long (say, half-hour) scenes and combats, others have multiple storylines/scenes going on at once.
• Some games feature a tactile component, some don't.
• Some tables are happy with first-person improvisational scenes playing out in unexpected ways, others want to stick with third-person or "zoomed-out" scenes.
• You're almost certainly in a public location with other games going on, foot traffic, and lots of noise from other activities.

In no particular order, here are suggestions for how to turn some of those things to your advantage, how to ameliorate others, and how to make use of your player's strengths. All borne from personal experience.

1. Use minis/markers/scrap-paper chits. Even if you'll never sketch out a map, have them out on the table to represent marching order, chatting with NPCs, exploring town. I keep a miniature croupier's stick in my game-bag and this, along with minis on the table that need to be moved every ten minutes, probably occupies ~30% of my fidget-time. This requires some self-control, as they can't be constantly playing with the minis in the middle of the table. But even during social scenes it can help to be all "So, Princess Vespa. At last I have you in my clutches."

2. Ask them to take good notes. It's a second set of ears to catch a detail you made up on the fly. It's a virtually-constant mechanical task if they embrace it. Flipping back through notes to find a piece of information is another side-task they can call on when needed. And you've got copious campaign notes. Where's the downside? This captures ~60% of my fidgetry.

3. Encourage them to sketch/doodle. The entrance to the temple you just painstakingly described. The layout of the buildings you mentioned in this village. Their character's signature spell going off. This occupies the last !10% of my fidgetry.

4. Have them shoulder some game-responsibility. Got 20 goblins' HP to track? This is your player-assistant: they can play their character and keep an ear out for damage done to mooks for you. That sort of regular task-switching actually eases their experience, and it (hopefully) relieves some of your game-running burden. This obviously requires some trust in their honest play, which seems at question. But I'm not sure how to trace that back to ADHD. I think you need to address that separately.
5. Use them as your reference librarian. My AL GM rarely has to look anything up, because I've already got the book open to the page they want. Again, we've ticked up some quick task-switching and tactile engagement, search-and-find simplicity, all hopefully making your life easier. Again, there's some self-control needed here: I always lay it out there for my GM without saying a word, and just close it up and put it away if he doesn't reach for it.

At this point, btw, we should also have solved the "I have to tell him the scene four times" and "he's on the phone" problems. He's likely on the phone because passively listening to something cannot occupy his attention. But "listen and jot the name and one-line description of every NPC, occasionally flipping to a little-used spell's description" may just be the sweet spot for attention and engagement.

6. Choose (your seat) wisely. You're in a super-distracting environment. Pick a corner/side table if you can. It may be tempting to seat this player with their back to the room--resist that. Seat them by the wall, facing everything. IME the visual distractions aren't the problem as much as the conversations at other tables. It's much easier to avert your eyes than it is to avert your ears. This, of course, helps your other players, too. At an AL location with more than two tables or multiple events going on, everyone's working overtime to filter stimuli. This also probably requires someone--likely you--to be good abut getting there on the early side. That's your talk-with-the-tough-player time: ask them to also come on the early side.

7. Schedule breaks. I don't know about your AL, but my sessions often run three-plus hours. I have a player at one site who really cannot sit for more than twenty minutes--your player may be in the same boat. (By the way, it's probably undesirable for all the other players, too.) Announcing and maintaining a practice of drink-break 1.5 hrs in, bio-break 2.5 hrs in can do a lot. If nothing else, it means that during those two times the player's up-and-about they're not missing new activity. But it's not nothing: knowing that there's a scheduled break at minute 90 can make it much easier not to jump out of one's seat during minutes 70-110.

8. Discuss length of scenes, beforehand if possible. "We're heading into a town you've never seen. Do you want to RP the getting-to-know-you-parts for the next half hour, or do you want to approach it some other way?" When your table's decided, you follow that decision. This way your player(s) who might have trouble focusing during those scenes have advance warning and they can bring their attention-management resources to bear.

9. Get in a habit of declaring intent when heading into a scene. A scene where the table's said "we're trying to buy access to the royal ball" does not sound like a good fit for some random-seeming blurted out idea coming from left field. But the scene "let's head to the docks and see what adventure we can roust" is a nice fit for a wild cannon. Get your table into the habit of declaring intent ahead of time, and thus let this (and all) players know when which playstyles are going to be appropriate or not. P.S. knowing your players' intent makes things much easier for you.

10. Give them explicit warnings. During some of your conversations be perfectly frank with them: "I can't have you disrupting play, it's my responsibility to make sure everyone gets a good experience. I will ask you to leave if I need to. But here's the other half of the deal: I'll let you know if that's coming. If it seems to me that you're disturbing someone else I'll tell you with a note. And I'll tell you if it happens again. And then I'll tell you to leave. And I won't hold a grudge." This, of course, is a fair way to treat any player. Discuss it with your site coordinator: they've got your back.

Final thoughts:

• You mention fudging rolls/cheating. It's hard to see how that would stem directly from ADHD, so I didn't really address it. I feel like it needs to be addressed independently, though to the extent that it might stem from boredom/frustration the above might help. This needs more-targeted intervention--the kind you'll find wisdom for if you look at question on cheating.
• You mention misusing class features. I've got a severe ADHD player in one group who regularly claims they've got features and spells five or ten levels out of reach. I simply say "no you don't," hand them the PHB open to the correct page, and skip over their turn in combat until they present an action which is within their character's abilities. (I check back quickly at the end of each other player's turn.) To be clear: this isn't a player trying to cheat, this is a player who--I think--had never read a paragraph of the class description front-to-back before we started.
• This player might be able to play in Adventurers League, they may not. No matter whether you help this player develop these skills or they end up getting perma-banned by the site coordinator, I think you'll end up the better for the effort. Keep up the good work.