I agree with C-Bears, you need either a Class A or 5th wheel. You should find plenty used models for either. Finding a nice 3-5 year old model may be difficult, but not impossible. If you go with a 5ver your tow vehicle will likely need to be diesel, not necessarily, but to have a unit that has a washer & dryer it's going to be at least 12,000-14,000 lbs. and a diesel is your best bet.
Good luck with your search. I think you're doing the right thing considering your career. Your pets will appreciate having you tow them around in "their" home.