It would seem like the quintessential Rationalist institution to congregate around is the prediction market. Status in the community has to be derived from a history of making good bets (PnL as a %, not in absolute terms). And the sense of community would come from (measurably) more rational people teaching (measurably) less rational people how to be more rational.

The founder of LessWrong / The Rationalist movement would absolutely agree with you here, and has written numerous fanfics about a hypothetical alien society ("Dath Ilan") where those are fairly central.