I remember someone with very similar aims asked much the same question a year or so ago. The general consensus, as I remember things, was that a dedicated animal transport trailer would be the most satisfactory system. Having several dogs (who may not know each other or you all that well) cooped up together inside a vehicle while driving doesn't sound to me like a great situation. Further, very many campgrounds limit the number of animals you can have per campsite, so instead of sneaking dogs into hotel rooms you'd quite possibly be sneaking them into campgrounds.
That said, I would look for a used RV rather than a new one, partly because the depreciation the first few years is rather staggering and so the cost would be a lot less. If you're looking at a class C, some unit with sufficient space and weight carrying capacity most commonly would be on a Ford E-450 platform, and have their gas V10 engine, assuming it's not older than about 18 years or so. There's nothing wrong with that chassis combination; it's solid and well-known and generally reliable, though not particularly modern in overall design. It basically is an enlarged E-series van chassis.
The most important thing to check in my opinion (besides whether the layout and general design of the RV would work for your uses) is condition, and condition largely boils down to not having leaks at the seams or things that must perforate the roof or walls, and having the various systems work (furnace, AC, generator, plumbing, fridge, etc.) I would much rather have a well-maintained older, lesser quality brand RV than a newer, better, but unmaintained one.
Do be aware that an RV requires a good bit of maintenance. You will need to spend time and money keeping it in good shape if you do get one. The gas milage will also be noticeably worse than with a van (maybe 8 mpg, give or take a bit, for many class C motorhomes).