A small freezer chest in the garage, basement, other non-eating room is your best option. Could you get gift cards, in advance, for Chanukah? That would help you save up quickly. My son bought me my chest freezer and told me not to even think about Chanukah, my birthday, or Mother's Day for a full year. Would someone do that for you?
The only other thought that I have is to separate out the bottom level of your freezer if it is a side-by-side, or one small corner of the freezer if it is an on-top. Use very thick cardboard or heavy plastic to make a separate area within the freezer. It can't be used for anything else, and when you got your own freezer, this freezer would have to be defrosted and re-kashered.
This sort of skirting the law will not be acceptable in a Hasidic or Orthodox household. Conservative would depend. Reform or Reconstructionist would be fine in almost all cases.
or...how about using a tiny bit of space at a nearby friend's house? You could keep your couple of packaged mice in a paper bag, tucked into a little Tupperware container.
Let me know what you decided to do. I'm interested. I'm Reform, by the way.