Erik Schmidt's answer is probably the better one to go with (as it'll help you find the root cause), but I'll contribute a bit based on what I see from your description.
From your description, you have a player who enjoys:
And who doesn't enjoy:
And yet, this player is playing in a campaign that is low combat, high intrigue, and where the penalties for building a new character are incredibly prohibitive.
When players are marginalized, they tend to try to get into the spotlight in any way that they can. And that leads to disruptive behavior.
This leaves you with a few choices:
Try to get the player into a campaign that they'd enjoy (either by running an additional campaign, or by encouraging them to find/create a new group).
Remove the player from your campaign.
Find ways to compromise, attempting to bring your campaign and the problem player closer together.
The important thing to realize is that there is no way to force the player to like playing the way you and your other friends are playing.
Character Jumping
Some people like to play the same character, and enjoy the process of growing and developing that character. Other players prefer the creation process, authoring new characters and mechanical combinations.
Dealing with the first sort of player is pretty easy. That's traditional, long-form writing. We see examples of it all the time in many different forms of fiction.
The second sort of player, the one you have now, is a bit harder to deal with. Here are a few things I've picked up to help deal with that:
Avoid introductions
Either introduce the character during downtime ("And so the party rested for six months, and met a new companion"), or use the "poof" method ("of course I've always been a catfolk fighter. Human wizard? Ridiculous."). You want to avoid having players stuck constantly in the untrustworthy rookie slot.
Maintain player parity
New characters should be (or quickly become) the equals of established characters. It's no fun being the one guy who doesn't have an artifact.
The rule of thumb is that anything that's relevant frequently, whether it's powerful mechanically or narratively. A new character can probably be allowed to do without a one-time favor from a noble. But they should probably get written into a life-debt from the king of the current kingdom (or be given something else).
If the new character doesn't start with this stuff, then the stated goal should be to catch them up quickly (within a few sessions).
A level penalty is right out, unless it's a temporary one (or you're playing a game where levels aren't super important). For example, in my 7th Sea campaign, new characters start slightly below the lowest player. But all players are brought to the same experience total periodically.
Let them know that they won't get some of the depth other characters do, but make an effort anyway
From a DM's standpoint, the hardest thing about a character jumper is that you can't be sure of who they will be in the future. So if it takes you three months to author a storyline featuring that character, and they change characters every two, that creates an obvious conflict.
Let the player know about this, and give them an opportunity to work with you. And make an effort to work their current character into the plot from time to time anyway, as a show of good faith.
Combat and Intrigue
It sounds like this player enjoys combat quite a bit. Most likely, it's a chance to see how the mechanics of their character work, and to show off a little bit. This is all fine, except when it stomps all over the intrigue that other players want.
Give them combat
The easy answer here, is to make sure you're providing combat. Don't make talking the ultimate super-power. Give him a mixture of fodder that he can easily crush, and challenging mechanical opponents.
This is where the bit about parity and artifacts above becomes important. If their well-honed killing machine is constantly shown up by the senior characters with artifacts, the combat probably won't scratch their itch for them.
Signal shifts to intrigue mode
Provide clear signals to the group when a shift to intrigue mode is happening. Set intrigue in populated areas (cities, etc.) and combat outside of it. Have combat-based foes rush the group, snarling (or ambush them with a poisoned blade), while intrigue-based foes sit calmly on their thrones.
If nothing else, the occasional reminder that "Baron VonEvil is pretty well connected. Are you sure you want to fireball his face?" can help quite a bit.
Always have a backup plan
This player is going to blow stuff up. As the person who's planning things, that should be a possibility that you consider. How do you keep the plot moving if the villain doesn't get a chance to monologue? What happens if the whole room is fireballed?
The goal isn't to always have a way for the PCs to succeed per se, but simply for the story to always have a way forward. Perhaps that means that a scheme of the late baron crops up shortly after the heroes fail to find the documents, and now they have a fresh chance to investigate.
Or perhaps the important documents were partially sheltered by the lockbox they were in. They've been badly charred, but there's enough there to make out a new location to investigate. Is it a trap? Of course it's a trap. They're going in blind.
Communication
Talk to your players. Not just about how you want them to behave, but about what you're trying to do, and how you're trying to shape the campaign to the individual players. Make it a dialogue, and solicit their feedback.
I'm trying to add in some more interesting "boss fights." What did you think of that last one?
I really liked the way you guys managed to outsmart Baron VonEvil. That's given me some interesting options for the next couple of adventures.
My goal is to tell an epic story of your rise to power. I think we could use a bit more of the political stuff, I just need to work out how to write that.
Etc.
Recommended Reading
There are two excellent sections in roleplaying books that I'd highly recommend for dealing with this problem, if you can get your hands on them.
The first is for the entire group. It's located in the 7th Sea Players' Guide, on page 238 under the heading Resisting the Story.
It's a couple pages long, so I won't reproduce the entire thing here. But the representative part of it is this:
Don't poison the PC pool. Don't act destructively just because "it's in your character," and don't force other players either to ignore your action or kill your character because it's in their characters. Know where the line is and make sure you -- and your Hero -- never cross it.
This doesn't mean that there shouldn't be any conflict within your party. On the contrary, some of the best and most rewarding roleplaying experiences come from moral or ethical disputes between Heroes. As long as you know where the limits are, and when engaging conflict becomes irreconcilable conflict, feel free to pursue arguments with your fellow Heroes.
The Rich Burlew essay Making the Tough Decisions is another valuable discussion of this principle.
The second is for the DM. It's located in the 4th Edition Dungeons & Dragons Dungeon Masters' Guide (pages 8 through 10). It's a breakdown of the different kinds of players, why they come to the table, and how a DM can target each one.
Summary
This is a lot of text, and only really brushes on a lot of important topics. The core assumption here is that you would rather play with this player, than necessarily play your campaign as it exists now.
The key is to give the problem player things that they enjoy. With good communication, a willingness to tune your campaign to the players, and a bit of compromise, the problem player may become a lot more tractable when their needs are met.
Define the Consequeces of Success and Failure Up Front
This answer addresses a very similar question. I think everything I said there applies equally here. In short: if you explicitly define the consequences of success and failure, players are less likely to misunderstand the information and run off doing some nonsense.
Let It Ride
In your case, there's one other trick I would add, shamelessly lifted from the RPG Burning Wheel:
A player shall test once against an obstacle and shall not roll again until conditions legitimately and drastically change. Neither GM nor player can call for a retest unless those conditions change. [The results of] the initial roll count for all applicable situations in play.
In other words, say you roll to search a room for stuff. You get an 8. That's it. That's your result for searching the room for stuff. If you retry the action, you don't reroll. If you try a new action, too bad, you're not gonna gain anything more. Likewise, if I roll 15 to climb a cliff, that applies to the whole climb; the GM can't ask me to reroll every 10 feet or anything stupid like that.
Let It Ride makes sure that rolls actually matter — you can't just turn a round an immediately invalidate something with another roll. Beyond that, it keeps the game moving forward. When we know that each and every result will stand, we can all focus on moving forward incorporating the result.
If you can't abide by a success or failure outcome, then don't put it on the table at all. Manufacturing excuses to reroll until you get the results you want is a sign that you need to rethink how you're scoping consequences. It's possible you shouldn't be asking for a roll at all.
If you want a situation to be a series of rolls, break it up into discrete tasks instead of just rolling amorphously a couple of times and then handwaving that, okay, now this one counts.
Call Them Out
If your players are constantly asking for rerolls, try just calling them out on their weaseling. Like, just straight-up say, "You're trying to weasel out of the outcome we already rolled for. Let's move on."
Why Are Your Rolling for "There is Nothing Here" Anyway?
If there's nothing to find, what's the roll about, anyway?
Occasionally there's some payoff to roll-to-find-out-if-you-know-that-nothing-is-here as a form of information-hiding, but from what I've seen, a lot of GM advice defaults to "Roll for everything just to create fake tension!" way, way too much.
In the example given, I'd only ask for a roll if I could frame it as something like one of these:
"If you succeed, you find everything of value in this room — secret passages, treasure, clues, everything." That way the PCs can discover something for their efforts even if it's not what they necessarily intended to find.
"Okay, so, time is of the essence, right? If you succeed, you find out right away whether there's a trap door here. If you fail, it'd take a long time to search properly." Now the roll is all about "What is the cost of the information you want?" I do this only when there is already pretty obvious pressure of some sort; otherwise you're just kinda manufacturing complications that don't really matter.
Otherwise I'd just tell 'em. There's very little down side to doing so. What's the point of trying to maintain a feeling of uncertainty here, unless you're trying to waste time on purpose?
Best Answer
Avoid the DM vs Player trap
It looks like you may have fallen into that trap, based on how you presented the problem. This isn't a matter of a "fair fight" between a monster (or a group of monsters) and a PC. It's about the player making choices that have costs. This player gave up more offensive power to be better at defense. Honor that choice.
That high armor class is neither a fluke nor an exploit. It is well within the limits of the game as built and comes with a cost: reduced damage output (and in the case of heavy armor, disadvantage on Stealth checks).
Your player took an opportunity cost to become hard-to-hit
Rather than taking the dueling style, which would have added +2 to damage for each hit, or Great Weapon Fighting, which would make his damage output higher, this player chose to be harder to hit. There is no reason for you to get frustrated - You Are Not The Monsters.
Monster defeat is not a sign of a bad DM.
The monsters1 will try to defeat the PCs based on your choices, to be sure, but the monster's defeat or difficulty in hitting a purposefully built hard-to-hit character is not a problem. It's a consequence of the players' choices.
Bottom Line: put yourself in a neutral mind set, be the referee.
Do this before play begins: commit yourself to being a referee2 rather than a "side" in a battle or a contest.
Don't over-identify with monster/NPC success or failure.
The PCs are the stars of this movie anyway. They are by design scheme intended to become larger than life. Take a look at these descriptions of the tiers of play in the Basic Rules, p. 12:
The dice are your tools, let them help you be fair
Let the dice / RNG remove your fears of metagaming. They are fickle and they are fair.
It doesn't matter if the monster hits or misses. Roll the dice and see what happens. That little rodent who rolls a 20 or a 19 still hits this "tanky" character.
Let RNG help you. If you feel that you are picking on one player too often, or that another player never seems to get attacked, use a random die roll to see which character gets attacked when more than one character is within range. I've been using that tool since I started DMing.
Notes:
1 For the purposes of this answer: Monster also applies to any NPC opposing the PC's.
2 It is worth noting that when D&D was first published in 1974, and before "dungeon master" became the term of art, the person running the game that the players were in was called "referee" or "judge." Both of those terms have the connotation of impartiality: embrace it. It'll make you a better DM.
RNG: common abbreviation for "random number generator"