I’d love to play or DM a Paladin in this scenario.
You do need to make clear what will make the Paladin fall. For example, I wouldn’t rule that any of the actions so far would warrant a fall in one of my games. As long as he is pursuing a conversion or redemption of the antipaladin, this plotline can be a lot of fun.
I also don’t agree with screwing over the paladin either. For my tastes, I’d pursue scenarios where the paladin has to thwart her plans, and may even have to fight her, but he’s fighting non-lethal, and she’s outright trying to kill him.
IF he is able to turn her to the path of righteousness, the conflicts do not end either. Her former allies may not take such an act kindly, or may want to stop her from revealing more secrets. They may send high level assassins to wipe out such an embarrassment.
The BBEG may also have loved ones or items that are important to the anti-paladin to ensure her loyalty. If she were to betray him, a loved one may be targeted for death as well. So before she can leave her current organization, the paladin would need to secure the safety of such loved ones.