Bit of a background: my group is mostly new players, and I am a new-ish DM (only having DM'd one other game before, and it wasn't even a DnD system). To make things simple for myself and the others, we run a very low-RP game type. Players are free to create backstories and character arcs if they want, but usually, we keep the strategy simple:

1) go to place for quest
2) kill any bad guys in way of quest
3) get paid for completing quest

We do have some other quests, like guarding a VIP and using diplomacy to unite allies against an enemy, but we still don't run a very RP-heavy game, and my players are still having fun with it.

Recently, however, we got a new player. His character comes with a backstory, a personality separate from his own, and actively engages with the story rather than killing everything. He actually BRIBED an enemy encounter at one point. He even - and I kid you not - wore an actual rogue's hood and brought a realistic dagger to the table with him for authenticity (granted the robes and dagger made little sense considering he was a warlock, but still). I am about to give this guy a round of applause for his commitment, but then he hits the table, aaaaaand....
He never stops roleplaying and hogging the spotlight. Shouting over people, taking charge when not needed, backseat gaming what others should do, you name it.

Now I am in a pickle: should I ask him to tone down the roleplaying and stomp on his fun? Or should I provide more opportunities for him to roleplay and hog the spotlight, making the other players enjoy it less?

Any suggestions?

There's a fine line between good RP and being a wangrod. This sounds like an "away from the table" discussion, but it shouldn't be too hard. But first of all, how do the other players feel? have you talked to them? Do they feel like he's always hogging the spotlight and ruining their fun, or is that just how it appears to you as an independent observer? Talk to your players. Now, if they feel the same way. Talk to the player.

Tell them you love their RP. You think it's great, and they bring a unique element to the table. Then also tell them that they need to let other players have their turn, that they need to stop talking over the top of people or telling them what they should be doing. They'll probably say that "its what my character would do". There are a million threads about that kind of comment. But generally it boils down to them being a wangrod to quote Colville. Explain to them that D&D is a social game, where players need to enjoy playing with the other players. And that if their PC is such an obnoxious tool, then the other PC's would simply choose not to go adventuring with them, risking life and limb with someone they don't like/trust. And that the other players are in no way obligated to rescue/help/resurrect the PC if they were to go down.

And secondly, stop them at table. If they start yelling over the top of someone, ask them politely to stop. If they start telling someone what to do on their turn, tell them that its not their turn.

RP is great. Reward it with Inspiration or something.

Shouting people down is not cool. You need to step in as DM and impose some structure. If necessary, impose the rule that you can only talk during your turn in the combat order, that kind of thing.

It sounds like you have a more experienced player who's used to playing with more experienced players and maybe is taking charge to try and guide the newer players. I can def respect that, but yeah if that's just not the game you guys wanna play then I'd just simply, in a non-accusatory way, say "yeah that's just not how we like to play, wanna tone it down and kill stuff with us?"

Maybe later down the road you can see if they still wanna do something more RP intensive that everyone can try on, so everybody gets a chance to do the thing they want. 

I literally just had to deal with a similar situation and to echo the above this was my approach.

1. Spoke to all other players first individually, told them how I felt and asked for their input.

2. Spoke to the problem player on their own, made it clear that this was not a me vs them or group vs them but that their play was effecting other players and my enjoyment of the sessions.

3. Laid out what needed to change to move forward - this is often over looked in advice of just talk to them, but if you are not clear about what the next steps are then you leave it to the player to decide how they need to change. What if they don’t change enough, what if they are differently problematic? Part of the reason you are where you are is that the players decision on how to play wasn’t great, don’t assume they will get it right this time. So be clear about what you need from them going forward.

A thing I highlighted to my player is that talking to him and the other players takes time, it takes prep to handle tactfully and time in seeking help and advice from others. Which is time that could be spent planning the D&D sessions, creating interesting NPC’s, sourcing cool maps etc.

one thing I think that gets overlooked is that the game is supposed to be fun for all players, including the DM. Sure our role is more admin heavy and we put in 4 hrs work for every 1 the players do, but you need to be enjoying it. Maybe that enjoyment is running the session, maybe it’s the creative buzz you get from planning, maybe it’s finding out about all the lore? It’s probably not planning interventions for one player (unless it is, no kink shame here), that will drag quickly and lead to burning out as a DM.

the player may be having the time of their life RPing but the equation is simple “one player enjoying their RP < everyone being able to enjoy the game”

Up to the point that he was shouting over people, and backseat gaming, I say this is just a play style issue. I’ll touch on that in a sec, but shouting over people and back seat gaming can and should be addressed. I’m willing to bet he doesn’t even realize he’s doing it.  He’s just getting so into the game that he probably thinks of those things as cheerleading more than controlling.  And he might say as much during confrontation.  A good response from you might be “I really do get that.  I’m glad you’re invested in what I’m working to set up for you.  These other guys, though, need more space to invest in the game for themselves.  We’re good. Just be conscious of when you’re starting to talk over people or help them optimize their decisions.  The best memories of games is usually not when you executed a smart plan to perfection; it’s when a suboptimal plan, or no plan plays out for good or bad with results no one, even the DM went into the situation expecting.

As for play style, there’s a spectrum from RP heavy (feels like ‘Let’s play pretend!” to the other end of the spectrum) to combat focused (Feels like ‘I press the attack button until I get treasure’ to the other end of the spectrum.) Each can result in a good time, but not at the same time.  You need to be near your party mates on the spectrum, or you don’t feel like your mates have your back.  That alone is stressful sitting at the table together, even before you add in shouting over or back seat playing.  If both sides can adjust their expectations and style to move within reach of each other on the spectrum, that can fix it, but it’s hard to get people to do that, and it’s often easier to find another group that better suits your style.

Once you get past the intervention phase, and - hopefully - get the problem behavior under control ( no matter what style game you're running, "hogging the spotlight, shouting over people, taking charge when not needed, [and] backseat gaming" are problematic behaviors ), it sounds like you should have a(nother?) Session Zero.

I think that as a group you need to decide what kind of play style everyone wants. I note that you said "to make things simple for myself and the others, we run...", but was this communicated clearly to the Players that this was the play style? Are you sure everyone ( other than this one disruptive Player ) are on the same page? Clearly someone didn't get the memo. Is this approach really something your Players are asking for, or is this a choice that you as the DM have made to make your life easier? Btw, It's not wrong, or a bad choice, if you're doing that to ease yourself into the game as a "new-ish DM".

There's no reasonable style of play that is objectively wrong. However, it's absurdly easy to get people at the table whose preferences for style of play is incompatible. You see many DMs complaining about Players just wanting to be murder hobos, and here you are complaining that you have a Player who doesn't want just be a murder hobo :D It's all about compatibility.

I think you need to be clear what the wants are from each of the Players, and what your wants and comfort levels are as a DM. Then find a middle ground where everyone is content. Or recognize that a middle ground doesn't exist, in which case maybe not everyone can stay in the group, and some people have to be eased out gently, and without recriminations or rancor. Or recognize that you need to "work up" to that middle ground and that everyone needs to be patient until you get there, if - for example - all your Players really want to "roleplay instead of kill everything" ( just an example ), and you need to work your way up until you're comfortable running a game like that. Or maybe you're extraordinarily lucky, and you're already there, except for that one Player.

But at least then you know rather than just assuming.

Deal with the disruption, stop and take stock of what everyone wants the game to be, see if a compromise position is attainable ( perhaps eventually, if not immediately ), and see how you can get from where you are, to where you want to be ( assuming any change is needed at all ).

And have fun with it! Best of luck :)

P.S. Personally, I would have loved it, if some of my Players were smart enough to bypass an encounter by unusual and clever tactics like bribing their way out of the situation ;)