He sounds like an exhausted and confused little 2 year old. Poor guy.
I would actually start by insisting that the 1:30 drop offs cease. Smack dab in the middle of naptime, or for that matter anytime after nap begins, would NOT work. And if its an issue of her not wanting to pay for that hour, I would actually offer it without charging, in order to facilitate our scheduling. But it wouldn't be negotiable.
I agree that the only way to enforce that naptime needs to be a specific way is to give up about a weeks worth of your own down time. As playfelt suggested, just keep going in, laying him back down again, and getting him to settle. I wouldn't leave any toys or stuffies, etc. Just his blankets. If he's climbing out, but you feel that he should still be in the playpen, then place him back in and use your stern voice. "NO climbing. It's naptime." Then walk away and watch him. As soon as he gets up, go lay him down again. It's an arduous process, but so very worth it.
As for the overall problem of him having no definable schedule and a bad sleep pattern; it depends on how far you are willing to take it with his mom/dad. I would let them know that you feel like the constantly changing schedule is, indeed, impacting him negatively. Ask if they would be open to working together to develop some consistency. Beyond that, you can't make them do it. If they aren't willing to work on it, you need to decide if you can continue as is, or if you're willing to work on it solo.