There is a need for a written agreement for clarity and because, typically, the adults involved do not get along well or trust each other with spontaneity: this is why they divorced in the first place. In this situation OP reports begging the father for his involvement, but he seems to be occupied instead with getting in sarcastic, cheap shot digs at her instead.

In this situation OP has offered what strikes me as a typical propoal for such situations. Maybe, from her ex's perspective, there is room for improvement in it. Her ex is free to offer a counterproposal such as "i would rather have her every other full weekend, not one day per weekend" or "Could you live with 24 hours notice?"

Givrn that the ex seems to be unwilling or unable to work directly with OP, perhaps a mediation service would be helpful, as suggested by Crazy Hitch.