buy travel trailer or class C ~~ with multi small dogs

Many national parks are not particularly dog friendly. Many campgrounds, both public and private, have limits on the number of dogs (often two). These two things may limit your choices and destinations a good bit...at the least, you'll have to do a fair amount of research to find places to stay.

Look up "recreational towing" in the owner's manual for your Prius to see if and how it can be towed behind a motorhome. I think it may require a tow dolly, which is somewhat of a pain to use. (Essentially any front wheel drive vehicle can be towed with a dolly, assuming it physically fits on it.)

I would think a truck would get rather cramped on log drives with four people and five little dogs, and for that reason a motorhome might be more convenient.