You're baiting them both, you know that. You aren't afraid of Batman and maybe you should be, but he doesn't care about you. You're not a threat, even if you did shoot Joker. You're not telling anyone about why you did that, and you're not even going to think about it! You are, however, going to get that cat out of the Greenhouse. If she's informing to the Bat, she's dangerous and you aren't going to let her put Ivy in danger. Despite the differences the two of you have, the arguments because you're not the right one, you care about her.
The truth of it is, you need someone else to guide you, to reign you in. You think Ivy might be less dangerous than the Joker. He fascinates you, there is no denying that, but she is so convinced that he will be bad for you. Everyone is so intent on keeping you away from him and although you're sure you know why, there's a part of you that you can't really trust what they have to say. You want to figure it out for yourself and maybe it's because you know she'll be there to take care of you if you fuck it up. Still, you'll get that damn cat out of the greenhouse if it's the last thing you do, even if it means goading the bat the come after you. You'll take it. You'll deal with it. Whatever it is, and maybe Ivy's just taking his place, but you don't care. As long as the cat is gone.