Bwhaaaa, oh man that made me laugh.
Like Nemo I’d go with baseball hitter. It’s a sport that does two things really well, honor statistics and remember the contributions of the greats for a very long time. There’s the thrill of batting in clutch situations where you ‘save’ the team and while I’m not normally a very ‘center of attention’ guy, I could get used to the raucous accolades from the announcers and crowds as I rounded the bases after a towering blast.
Plus it pays good.
There’s an enormous difference between those two descriptions - which is the first reason why I objected.
Next let’s take a look at your examples:
Morphy became World Champion at 22 and later didn’t need to work (because his family was wealthy.) No evidence of mental illness.
Steinitz was World Champion, published seminal books on chess (especially defence) and only fell ill at 60 when he was in poverty (and may have had syphilis;)
Alekhine was an alcoholic and married 3 times - and this makes him mentally unhinged? 
Tal was an alcoholic - so what?
I agree Fischer had a mental breakdown at the end of his life.
You don’t mention:
So your ‘mentally unhinged’ allegation rests on Bobby Fischer and Steinitz.
Almost all of the above in my list have received distinguished recognition in their own countries - and I’m confident the last four have all became millionaires as well.
Additionally, a 3% improvement over the previous best is huge. 3% better than Babe Ruth is:
.372 AVG, .504 OBP (!), .720 SLG, 1.224 OPS
And those are from his career averages, not just his prime. You’d be in charge of maintaining yourself, but that’s not really a Ruthian challenge…wait…
I’d choose to be 3% better at soccer than Pele or Maradona (both adjusted for the speed of today’s game).
Pianist. I consider it a technical skill, whereas singing is more subjective.
And please, no pianist jokes. 
Piano playing isn’t really a competitive sport, though, is it?
Unfortunately, I’m going to have to go with golf, a sport I hate. Actually, I’m not much for sports in general, and am completely uncompetitive, but what the hell. Hopefully by being awesome at golf for the past 14 years (since I was 18, right?) I will have accumulated enough money to retire in 6 months and live off the interest. Also, my body won’t be freakishly muscular and I won’t have destroyed any joints. Acceptable.
I would go with something in the horseback riding area, but I don’t think the money is really there, and I don’t want to have to keep this up long term.
Rock-paper-scissors.
I’m going to say no… fields like science or being-a-chef or being-a-doctor probably do have some competitions you could enter but are not primarily ones in which you engage in competitions with winners and losers as the main thing you do in that field.
Obviously this is just a silly hypothetical, but… I didn’t intend to throw in a free fountain of youthing along with it. Feel free to answer as either “assume you are 18 right now…” or “assume this happened when you were 18…”.
I originally thought baseball due to injury concerns with other sports, but fuck it: I’d want to go down in Canton as one of the best QB’s in NFL football ever.
I’d pick whichever would let me score the biggest windfall in that 6-month grace period (subject to the constraint that it’s not something with a significant chance of wrecking my bodily well-being), then go back to not bothering with whatever it was and sink into obscurity. I don’t know precisely what that would be, but golf and e-sports are both probably pretty good contenders.
It’s a competitive activity, certainly. Nobody said sport, you just assumed it.
But how do you quantify a 3% improvement? What’s the (ha!) scale you’re judging by?
Same problem with things like gymnastics, diving or figure skating.
Poker is a good one – I’d like to sock away a bunch of cheddar in 6 months, then either keep playing in celebrity tournaments with starlets and stuff and try to learn how to play, or quit.
But who’s going to stakehorse you over the Internet? Or IRL? I guess that’s solved by the retroactive memory fix, so still poker is it. Spend eight hour days dicking around online for six months, make a shitload, then do whatever.
If the retroactive memory wasn’t there, and stakehorsing were a problem (no one in their right mind would loan me enough to last in a high-stakes series of games), I’d choose pool hustling. Not tournaments, necessarily, because I don’t think they pay very much. But money games in bars and the few poolrooms left.
Or baseball slugger. Maybe baseball – secure paycheck. Not as cool – literally, you have to spend your days in the sun, not in a dark bar, having some whiskey or beer. Definitely quit after 6 months, or use my reputation to get a big money contract doing the least amount of work possible. Costanza’s job, maybe, but with more money.
Unless you start out utterly flat broke it shouldn’t take you THAT long to run $100 up and up and up and up and up if you’re literally the best poker player who has ever lived.
Would high-level finance be considered competitive? It’s definitely measurable and has a strong competitive point - it’s just that the game never ends. Being the best investor on the planet, even for six months, could be an extremely lucrative position.
Definitely not.