It's true that many campgrounds prohibit leaving a pet unattended, but most of those are geared towards short term "recreational" use. Public facilites are a good example, but you likely won't be using those much anyway since nearly all have length-of-stay limits too.
On the other hand, leaving a pet unattended at a "long term"-type RV park is likely no more problematic than doing so at a motel or other housing like you're presently using. You just have to shop around. And since all such facilities have power, keeping the dog warm/cool as necessary should be no problem.
It's really sorta hard to say though if this will prove to be a money saving thing- buying and maintaining a decent trailer doesn't come cheap, and neither does parking one, even by the month. Guess it depends on how those expenses compare to whatever your rent is costing you right now.
