Peter,
I think this would be a bad idea on multiple fronts.
Leave the trailer unattended at a rest stop? Not unless you’re trying to get rid of it!
I’m not sure how many schools would allow you to camp there over a weekend. People in the town might not appreciate it.
I also wouldn’t want to tow a trailer to VT in the snow. It could be tricky enough with just a car or truck. I’ve been in VT when they’ve closed the road due to bad conditions.
Also, it can be COLD! I’ve been in VT many times when it is below 0F. No way I’d camp in that.
Seriously, a lot of places offer ski and stay packages for almost the same price as a lift ticket.