Different places have different rules. Basically, most USFS, COE, and BLM areas have few restrictions on dogs except 6ft leash (some places will allow dogs off-leash if under voice control, but be sure your dog will be safe first).
National parks and state parks are more likely to limit dogs to the campground and paved roads only (no hiking on trails, no beach, etc.).
Many places do not allow dogs to be left unattended in the campsite. This may or may not include leaving them in the RV.
Barking is a big issue, so you want to find out if your dog will bark or not BEFORE you start leaving it in the trailer alone. You can do some trial trips at dog-friendly campgrounds. Ask your camping neighbors how your dog behaved while you walk around the campground, etc.
The other big issue is temperature inside the RV. If the weather is hot, you may need to run the A/C. Are there limits to generator use during the day? Will you have electric hookups? What if the generator died halfway through your outing and the dog goes without a/c for 2+ hours? And so on.