I'm Imperial. I don't want a peace treaty. I want to kick Ulfric's butt up around his ears. But I asked Balgruuf about trapping a dragon, not realizing that this would trigger the start of the Season Unending quest, so here I am.

Can I just go ahead and settle the civil war using the General Sherman Method (I'm at the attacking Fort Kastav stage) or will that break everything?

The treaty can be discontinued whenever you want, it's pretty much meaningless. Just continue the civil war questline after you're done with season unending.