tl;dr my mom is moving and she can only take one cat with her

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tl;dr my mom is moving and she can only take one cat with her and she's going to put down the other but no one wants that. what do?

my mother is moving in with her mother for around a year until she gets her own place, and her mother is only allowing her to keep one of the two cats, and she's going to put the other down. my siblings and i are very attached to this particular cat and none of us want to see it go. my dad won't take it in because of dumb reasons and i can't convince him to take it in. my mother says she doesn't want to put it in the shelter and see it suffer. her mom says the only way she'd take in two cats is if i also moved in but i don't want to be a burden and i wouldn't be able to until late december anyway. none of my close friends can take it in and it's too late to ask any other friends because both cats are 400 miles away from home. i am too and i'm going back tomorrow morning so i only have a small window of time to save this cat.

what do i do about this?