I don't think a towed RV will work for you (TT or 5th wheel). You're going to need to be self contained to keep the climate controlled in the RV. There are 5th wheels that have a generator but, I wouldn't trust that for hours at a time with no checks and balances. I have seen people add generators to travel trailers but again, I would feel uncomfortable with that solution. The simple fix is a working generator and roof air in the same space that you're occupying. That way you know if something stops working and can take some kind of action. The RV generators, typically Onan, are well made and will last a very long time if serviced properly. My 1990 Dometic AC still blows ice cold.
Overnighting at a Walmart is doable as long as you plan your stops and be sure that the Walmart in question is cool with overnighting. Some of them won't allow it. Also, some campgrounds might accommodate you for a night as long as the neighbors aren't negatively impacted by barking dogs and poop.
We have been RVing with dogs since 1988. Never once have we had an issue. The key is to consider your neighbors peace and quiet. Get a couple of x-pens but, don't just put the dogs out there and forget them. If we intend to give the dogs a lot of outdoor time we leash them and go for a walk. Our pugs are indoor dogs so, a short walk around the campground and they are ready to chill out and watch some TV.