Maybe it's a bit easier for me since I'm 24, but I just went out and got my fish. And my mom just sort of washed her hands of it and gave me crap for leaving a mess in the bathroom when cleaning the tank. But when the fish got here, she loved him. I don't recommend that for everybody, especially since your mom will have to take care of your fish when you're at your dad's house. It's highly likely that the move is totally stressing her out, so you may just have to wait till after the move.
But assuming you really don't want to wait, I might write her a letter explaining why you want a new fish, what you like about having a fish, and telling her that you've been feeling sad and lonely. When you're a teenager, hell, even when you're my age, your parents might tend to think that you're just being dramatic. But if you take the time to sit down and write her a well put together letter/essay, without things getting super emotional, it might work. Just make sure you make it all about the positives and how you feel like you have a companion in your fish and you miss having that in your life. Stuff like that. And make sure it's all true, no extra fluff. Parents can see straight through that.
Other than that, if none of that works, I say wait till after you move. Because the problem probably isn't the fish. It's probably the stress with the idea of one more thing your mom has to do, and the stress is presenting this way.
Best desk buddy of Dorian, the Immortal Gyarados. Blue Mustard OHM in a 5 gallon tank.