I've got an answer for your problem, which I will address first, and then something for you. Yay!
Group Effort
This is the one avenue it looks like you haven't taken. You know you haven't taken it, and it seems you're reluctant to do so. Likely because, even if you did get the others behind you, this person would act out, become sullen, and maybe increase the drama.
Talk to each one together or individually. Be honest about your intent, who you've talked with, and what you want to accomplish. Stress that if they can't work together to help even things out, you might have to end things. It's not worth suffering in your hobby, at least not this much. GMing is suffering enough, having to juggle the drama is not worth it.
This is the step you need to take, now, before you have to...
Kick Them Out
Let's go over what seems to be obvious from your question. It's obvious:
- Your play styles don't mesh
- Your standard approaches to these situations didn't work
- Nobody is willing to do what needs to be done
Just because you are friends doesn't mean you should sacrifice your mutual fun and enjoyment in order to accommodate this person. It's perfectly reasonable to be in a situation where you all want to do thing, but one person doesn't want to do it the way the rest of you do. Trying to work around it is getting you nowhere and is ending up just costing you fun.
I don't mean that you have to be mean. But it sounds like the only way this is going to be fun is to part ways in this thing. I'm sure there are other things you can do together you find fun. I know, you said:
I can't really do because of the group of friends
But if you want the problem fixed, that's where you're at. Or, you can keep going with the status quo and keep hoping something changes. I know it's hard, but part of making sure this hobby remains an enjoyable one is taking the steps needed to weed out the things that are making it less enjoyable. I personally hate weeding, but dang it, the grass sure does look better.
First, ask if it is really a problem. Are the rest of the players bored watching a one-man show, or are they amused and having fun watching this part?
If it really is a problem....
Call for a Roll
Shopping shouldn't take all session, and if the player has had a bit of the spotlight already that session, then there is no problem with saying,
DM: Alright, we need to move things along. You're trying to persuade the shop owner to give you a discount on what? What is the end result you're hoping for?
Warlock: The stakes, the wooden box and few bottles of potions.
DM: Great. Roll for persuation, with advantage.
**Warlock:**16.
DM: You talk at length and convice him to give you a 20 gp discount. Do you take it or leave it?
Don't always go this route, but sometimes you just have to move on.
While X, what are you doing?
Purposefully shift attention to the other players. It allows him to have his moment, but not loose
Warlock: Your stunning array of woodcraft is beautifully carved. How much would it take to buy one of these fine pieces?
DM: Okay, so the warlock is going to try sweet talk a deal with the shopkeeper. While he's doing that, what are you doing, Fighter?
You'll see Matt Mercer of Geek and Sundry's Critical Roll use this a lot when the party is shopping
NPC Shift Spotlight
Or even have the NPC make the shift.
Warlock: Your stunning array of woodcraft is beautifully carved. How much would it take...
DM[as NPC]: Yes, yes, sell your flattery elsewhere silver tounged worm. [to Dwarf] Do you hail from the kingdom under the mountain, Yes? So far from home. What dwarven work treasures do you have to trade?
Skill/Perception Interactions
Or if you have a player with high passive perception, pass that player a note while they're talking about something they notice, while the Warlock tries his thing.
You notice blood on the hem of the store keeper's tunic, and the shopkeeper is trying to hide it by holding his cloak close around him. As you watch him, something else seems off. Is his accent different than the last time you spoke with him? Where his eyes always that shade of gold? Something is just... off.
They can then choose what to do with the information. Maybe they let the warlock keep going to see what happens, but now they get to enjoy an aspect of it the Warlock doesn't know about. Maybe they will interrupt the Warlock, and try to get the party out of there. Maybe they will try to confront the shopkeep in the middle of the Warlock's corruption/manipulation/persuasion tactics. How fun for all involved if it played out like:
Warlock: Your stunning array of [DM slips not to Rogue] woodcraft is beautifully carved. How much would it take to buy one of these fine pieces? I'm sure I could pay a fair price of...
Rogue: I stab the shopkeeper while he's distracted.
DM: Roll for attack.
Warlock: Wait! What?
Best Answer
Don't focus on getting the more passive participant to be more active in the sessions.
I've had this experience before, and attempts to persuade the more passive players to interact more actively just caused those players stress and confusion as to the expectations. My experience as a teacher and as a DM corroborate each other: usually, people don't like to be put on the spot. If the passive player has not assumed a leadership role, you should not try to give them leadership duties, for example.
I don't think you have a problem from the perspective of the passive player. If they claim they are having fun, they probably are, even if they don't seem as engaged as the active player, or even if they don't show their feelings in the same way. In one of my groups that has been running for over a year, I have a player who is often very quiet, sometimes just observing the actions and dialog of the other players while doing her knitting, becoming more active during combat or if something directly relevant to her character occurs. And she absolutely does enjoy the game, even though I often think she's not having fun until she tells me later she had a lot of fun. So, if the player says they are enjoying it, I would not try to change how they are expected to play it.
Try to alleviate the leadership burden carried by the more active participant.
In your scenario, it sounds like the only real problem is that the more active player feels undue stress about having to push the game forward and fulfill leadership roles. Perhaps the stress would be alleviated if you railroaded them a bit more and/or provided an NPC who can fulfill some of the leadership roles. In general, this is not advice that people like to hear, and I often get shot down for even suggesting it. However, if you can take some of the burden of frequent decision-making and leadership off of the active player, whether that is by contrivance or an NPC, you may improve their play experience. And considering the frustrations of the active player, it sounds like they would actually appreciate having some of the decision-making taken from them.
Normally in a larger group of 4 or 5 players these active/passive dynamics would balance out on their own, but, since you only have 2 players, you yourself as the 3rd "player" (we often overlook that the DM is playing the game too) might have to act as the balancing force by giving more agency over the flow of the game to NPC's and other external forces.
Try to hook the passive player in ways that will not distract from the main adventure.
If the passive player occasionally pulls the group on tangents, first see if the tangents are unappreciated. If the more active player enjoys them, and if you enjoy them (but are merely worried about the group enjoyment), and if the passive player enjoys them, then perhaps you should let them be. But if the detours are unwelcome by you and by the active player then maybe you can redirect the passive player's focus slightly.
Find out the kinds of hooks that generally lead them off on the random deviations. Perhaps they are looking for some more lighthearted scenes or something to blow off steam during a more heavy plotline. If that is the case, sprinkle in more B-plots and C-plots to supplement the A-plot. Those B- and C-plots could be as simple as a single round of combat or a 15-minute roleplaying detour, but those small detours could vent the pressure that might be building up leading to those big detours that happen every once in a while.
As always, consider talking to the players first.
In this case, due to the size of the group, I would recommend you talk to each player individually about any changes you might consider making. Iterate on your ideas through those conversations until you figure out something that will work for everybody. Since there are only 2 people to talk to, talking to them separately is not that inconvenient, and talking to them separately should make them a little more candid and open about their thoughts.
Best wishes for your campaign.