There are some things you're forgetting in your analysis. For instance, Anak is going to be with her head on the chopping block regardless of what happens on the Name Hunt station. She is a child of a Zahard Princess and, as such, doomed to be persecuted until the day she dies, those who blindly enforce Zahard's law die, or she gets recognised as official Zahard Princess (or something).
Androssi indeed helped Baam a lot, but I already said that she's the only one (aside from Laure) with the right to expect something in return from Baam. The thing is, it doesn't even seem that Androssi had anything to do with Leesoo's plan, as it appears that she has been moving independently of Team Leesoo ever since she lost to Kaiser.
Also, the situation when Koon manipulated the would-be Team Leesoo is fundamentally different to what Leesoo has done. Leesoo didn't try to manipulate Baam, he outright decided to use him. Koon steered them toward the outcome he was hoping for with reverse psychology and peer pressure and hey fell for it easily, but Leesoo simply decided "this is what your role is". Moreover, Leesoo decided to bring up something Baam desperately tried to escape; something that--as a matter of fact--Baam had to suffer though partly because of these very people (i.e., FUG's hostages). In other words, Koon is well justified to send Leesoo to hell with his plan, and it has nothing to do with "being on the receiving end of manipulation", though I'm sure Koon would be quite annoyed by that as well.
Moreover, it should be noted that reconnecting Baam to his FUG identity is not unlike announcing openly that there is an illegitimate child of a Zahard Princess. Leesoo simply prioritised his own teammates over Baam. This isn't exactly wrong, but it clearly makes a distinction between those Leesoo cares about to protect and whom he's willing to throw under the bus to save them. Which also means that Baam and Koon are fully entitled to extend the same treatment to the Team Leesoo, because this works both ways.
As for the history, I don't recall Androssi risking her life for Baam--at least not more than she would have done without Baam in the picture--, though I do recall Baam shielding Androssi against an ignition weapon on 2F. Also, it appears that climbing with Anak is about as much a threat to Androssi's position as associating with Baam, so it's clearly not that definitive issue to her. Regardless, Androssi may have risked her position, but she has yet to sacrifice anything to help Baam, however, the current situation would require Baam to sacrifice some things in order to help Androssi (i.e., it isn't repaying in kind, it's grossly overpaying). Moreover, the game with Bia Princesses was clearly a setup to lure Androssi to the Name Hunt station and to get her trapped there; what would make anyone think that they plan on keeping (or even have the authority to keep) their end of the deal if Androssi really steals Kaiser's name? Androssi is going to be safe from the Name Hunt station regardless of whom steals Kaiser's name--be it Androssi or Baam--, so this isn't an issue.
Also, ever time Team Leesoo helped Baam was either because they were tricked (by Koon on 2F), or the help was only marginally balancing out the previous mess they'd caused (e.g., workshop battle help for pre-workshop battle assault). As for individuals, Anak has never helped Baam as far as I can remember, only made his life more difficult on 2F; Leesoo's help about balances out his assault on Baam; Hatsu has helped indirectly by being beside Androssi, and by being Lero-ro's "pawn"; and let's not even get to the likes of Parakewl. Laure and Androssi are really the only two currently on Team Leesoo who actually helped Baam, and whom would be justified in expecting Baam's help. And while Laure is clearly asleep, Androssi is asking for help in ch.205 independently of Leesoo's plan. Since they cannot both steal Kaiser's name, there will have to be some arrangement, and I'm sure there will be some options (e.g., Androssi joins Baam's team and becomes "one of those who stole Kaiser's name", etc.). Androssi phrased her proposition fairly strangely, but they will probably resolve this when they understand each other's circumstances and expectations. There is no indication that Baam has turned cold--even less so toward the likes of Androssi--, but he is likely going to consider the consequences and options more carefully from now one, because he realises that helping one often means hindering another.
In conclusion, Leesoo screwed up big time with this scheme, probably lost very important allies, and might actually make some enemies depending on how they handle Koon's rejection. It's even possible that the two teams undergo some member-exchange and go their separate ways after this is done. While Androssi's situation may still have a happy ending, because I doubt Baam would simply abandon her to her fate.
PS: That flip situation you describe is a false analogy. First, Karaka is one of the FUG Slayers, Ren is a RED nobody (i.e., youngest member, probably less than important). If this were an option, even some threat to Anak's life associated with such an exchange would probably yield everyone--including Anak and Baam--valuable credit with Zahard and the 10 Great Families for removing one of their principal enemies / threats. Second, free of FUG vs. free of the least threatening family Anak may have to worry about? Yeah, that's not really a choice that needs to be considered at length. Also, they are all FUG hostages, so they would be de facto saving themselves along with Baam. And this still assumes that they actually intended to keep their promise to Androssi, which is seriously questionable considering how Ren tried to play around with any Zahard Princess he encountered. Finally, as a bonus, unlike Baam's FUG identity, Anak clearly has pride in her identity, and instead of trying to escape it she is promoting it wherever she appears.