How big of a trailer?
When we were in the area back in 2019, I had planned on dispersed camping around Grassy Lake Reservoir between the two parks, but the road going into it was closed at that time. We ended up staying in one of the free park campsites along that same road ( 44.08565, -110.69886 ). It was nice to have a pit toilet, level ground and a bear box, but the mosquitos were the worst I've ever seen once evening rolled around.