Did Lando have to prove himself, though? Let’s say Lando did, and he brought in someone - great, he’s in. Now it’s her turn, but the point I’m trying to make clear is did she absolutely have to bring in Chewie? Clearly she could’ve brought in anyone else with a bounty if Lando was able to. Or, if Lando didn’t have to prove himself, and none of the other patrons do either, why should she have to? 😉
Lando probably proved himself in some other way. He was there and serviced Jabba for like, a year. He was needed to give Luke intel about things.
Lando is likely there so he can sneak the droids out after Leia frees Han. Luke wouldn’t just abandon R2 and 3PO (who were there to keep Luke’s lightsaber in case he needed it, bc he’d be searched upon entering if he needed to, and to deliver Luke’s message to attempt a non-violent solution).
JEDIT: Leia would bring in Chewie instead of anybody else first bc it’s less effort, second he’s on their side, he’s there to help if they need it (which, they end up needing it).