One thing that has confused me about Fukurou is how in chapter 322 page 7, Fukurou says that he is no longer a member of Kakerou. https://mangadex.org/chapter/713885/7 But during the fixed game with Suteguma he is a member again, as he gave his membership to Suteguma. https://mangadex.org/chapter/543875/11
https://mangadex.org/chapter/204672/8 In chapter 113 Vincent tells Caracal that he got a-hold of two Kakerou memberships, one for Caracal and one for himself. These memberships are never directly mentioned again I think and it's very unlikely that Caracal/Vincent used them for themselves, as Kakerou would know. And it's also very unlikely that that plot point was completely dropped https://mangadex.org/chapter/505675/1 Caracal most likely gave his to another IDEAL person, less suspicious than him, to use as a ploy to kidnap Kushinada.
So my theory at the moment is that Vincent was able to get into contact with Fukurou (some how) and make a deal with him for the membership, in exchange for him eventually transferring it to Suteguma during the fixed game. Something I just realized that may negate the "theory" is that Caracal asks Kushinada for info on all the members and my dumb-ass misread it as "referees" for so long, for some reason. So Fukurou may of regained his membership anytime between the game with Hal and that point of the story.
So if there isn't any contradictory evidence or something I forgot, it's possible that what Lalo "offered" Fukurou is his second Kakerou membership (but he had to give it to Suteguma eventually anyway...) But the "theory" does explain ,imo, your question of why "he was willing to throw his reputation for Suteguma and join Lalo in this gamble." And maybe it adds more to Fukurou and Vincent's relationship, idk.
But all I just said still doesn't really explain Fukurou's motives for allying with IDEAL in the first place, unless he is just "following his own nature," in your words. And he just wants to play games and Vincent offers him that.