With a $25k budget thats going to be pretty hard. Class A units in that range will be fairly old, and while there are some diamonds out there finding one used will likely mean 6 new tires and a couple of repairs immediately which make that budget line even tighter.
5er will need more truck than a 1500 Ram. Rams 1500's typically have low payload capacities which make almost all 5er's too heavy.
With the kids and dogs issues though you may want to explore another option. A travel trailer based Toy haulers "garage" gives you a multi-purpose area. Most have some form of fold down bunks and/or bench seating, and an easy area to contain the dogs when needed.
What you are trying to do sounds a little challenging, but it's another avenue to explore anyway.
On your other qx's
For health insurance, look to the states that have adopted the ACA, if you are going to be mobile you may fit the requirements for that, you would have to be willing to change your domicile.
On work while travelling, With you husband being a chef, look into management companies like Sodexo. The National Parks hire those companies to staff facilities in the summers for guest services, Yellowstone, Yosemite, etc have restaurant operations needing experienced chef' and kitchen managers and you may even be able to address the health insurance that way.
www.escapees.com has some great info on fulltiming and work camping, they may have some stuff that can help.
Good luck
