Does roleplaying inherently put you at a disadvantage?

Does anyone have any experience with roleplaying as a competitive player?

I do.
I'm so competitive I managed to win a game of Fiasco (a very non-competitive game).

Luckily, I know why you feel this way and where the source of the problem is.
Unfortunately, D&D 3.X is more often than not the cause of this dicothomy.

There's a thing game designers call reward cycle: encouraging the players to behave in a certain manner by giving them some mechanical advantage if/when they do.
This is done differently in different games where the authors are conscious about the need to encourage the intended behavior with positive reinforcement. Some examples:

• in the Solar System you get concepts that define your character called keys. following them often puts you into trouble or requires you to be in trouble. You get XP every time you hit a key.
• in Fate, whenever you do something that's not convenient to your character, you get awarded a fate point you can later spend to reroll, add modifiers to your dice or introduce elements into a scene.
• in Monsterhearts, you need to get strings on other characters to improve your chances of success. Things that get you strings are not nice things to do to other characters.
• In Primetime Adventures, players can reward other players when they describe a memorable scene by giving them a fanmail, which can be spent to draw more cards to play against the game master (whoever has more red card wins).

These games make bad choices for your character become good choices for you, encouraging you to play your character's flaws as well as his qualities.
D&D 3.5 also has a reward cycle, but it's not something the authors appear to have planned, unless you want to believe D&D authors actually wanted to encourage playing some psychopats who only care about money and xp.

• in D&D, you get experience points and gold for killing (or otherwise defeating) monsters, for good roleplaying or for plot-related goals.

Emphasis is because only killing/defeating/avoiding monsters has defined and quantified rules governing the gains. More than that, most of the things you can buy with money or XP are useful to be a better monsterslayer.

In previous editions, you gained XP based on the treasure you could get your hands on: avoiding confrontations meant suffering less HP (or character) losses. Now, the easy way to solve an encounter is "just kill everything". This means in a D&D game there's a straight path to "winning" the game that requires you to bash monsters in the most efficient way. The most efficient way, as many movie villains will tell you, is to get rid of all emotions and vulnerabilities. Weakness leads to being killed. Dying makes you lose some of the resources you gathered.
Of course, this does not produce a fiction that's satisfying to those who want to tell a "realistic" story, nor to those who want high fantasy, epic or similar results.

What usually happens in a D&D game where being a party of murderhoboes is not the intended result is that people is expected to behave consistently (and rolpelay an actual human being), and whoever can't find where the line lies and balance on it is usually either despised for "not being able to roleplay" or is willingly taking risks and worsening his chances at suceeding.
Someone calls it "role-playing vs. roll-playing".
It might be arguable that an equilibrium point or where none of the two happens exists.

So, what do you do?

• You can modify your reward cycle, by giving XP only for good roleplaying, but this often means "what the DM thinks is good roleplaying".
• You can remove this kind of reward. Leveling happens when the story calls for it, and money is given out accordingly. Then you introduce a different reward cycle.
• Maybe the easiest one, play a game that already has a "good" reward cycle. Buying your friends in could be hard, because of the investment needed to master D&D 3.5 and the fear that every system out there takes the same mastery to learn.
• Learn how to have fun without aiming at the intended reward cycle, while it's still there. If you manage to do this, please tell me how it's done.
• Just keep playing psychopaths (but that's not what you want, right?)

@Miniman wrote a comment where he suggested to roleplay a character who wants to make optimal choices himself. I suggest you don't: more often than not, being stuck on finding the optimal choice in-character is a flaw you'd probably try to instinctively avoid, especially when metaplaying is involved and you know a stupid decision of your character would make for a great strategy.
What you don't want to get is a character that behaves in a completely different way in different situations, because the lack of internal coherency hampers the kind of immersion you're currently looking for.