She would see that she has lost, while Tywin live and the Rock stands, she still believe there's a chance however small of winning, once we have taken Westeros, her side has already lost, which would mean for her to continue fighting us, wouldn't be trying to preserve her side, it would be trying to avenge it, and as long as we aren't too cruel to the Lannisters, I don't think Lanna is the type to go for vengeance.
This isn't anything special for the setting, except that instead of keeping her captured alive, we would be keeping her captured dead, it's the norm for captured warriors during a war, to be kept captured during the war, and then allowed to swear to the winner.
If you capture a knight in a normal war, you either ransom them or keep them captured, then once the war is over, they can swear to you without infringing on their honor, because their side has already surrendered.
This is how I see Lanna, she's loyal to the cause of the Lannisters, so as long as the Lannisters exist as our enemies she will be hard to recruit, once we have won the war, and the Lannisters have either been executed or stripped on their name, the way for her to be loyal to her family, will no longer be to oppose us, at that point the best thing she can do for them, is to either work for us, or work for someone we aren't at war against, and use the prestige from that to help her family make a new powerbase.
Of course we don't yet know that this is true, but we don't know whether she's opposing us out of hatred or duty either, if she hates us winning her over will be hard, if it's duty that binds her, once we have won her duty wont be to oppose us any longer, which make recruitment much more likely to work.