Would your schedule allow you to take on even Misty's feeding? Then your mom wouldn't have to do anything for her.

Could you sign up for some intensive behaviour sessions for Misty? If she could be trained to respond immediately to commands such as "stay", "sit", etc. (if she doesn't already), maybe your mom would find it easier to live with her. Also, some really cute tricks, done perfectly by Misty, might impress your mom and show her what a smart dog you have.

If you're really afraid Misty is going to be rehomed in spite of everything you try, do you have a friend nearby that could foster her for you for a couple of months, as a last resort? At least you'd know where she was, and could visit her and help care for her until you could get her back. I guess your dad can't take her, or you would have tried that already.

Good luck! I hope something works out for you and Misty.