Oh, it's an apartment in a house? Have you considered the possibility of getting a small shed? It would probably cost a few hundred dollars to get one big enough for a bike and trailer, but if you could get the space, that would probably be the ideal fix. I imagine an extension cord and a heater with a timer would really make it a nice setup!

EDIT: I should add that you'll probably want to keep the trailer as opposed to an on-bike kids seat, for two reasons. First, the enclosed trailer is a lot warmer and comfy for the kid when the weather is nasty, and on treacherous ice and slippery snow, the stable trailer would be preferable from a safety standpoint to the high-mounted on-bike seat that would really pile drive your kid into the ground were the bike to slip and fall over.