Go for it in the Class B.
Consider this; In the '80's my father-in-law took his wife and four other friends of hers, all adults to Alaska from Texas in a 24-26 foot Class-C towing a small Jeep. They had a wonderful time. Granted your Class B is smaller, but but you most likely have more "amenities" then his old Class C.
What do you have to lose? You are on an uncharted experience for you personally that will last you and the children the rest of their lives.
How long is your Class-B? in 1977 we were in a Chevy Sportvan 20 that was equipped by Travco. The top popped up and had canvas sides with zippered screened in vents. The van had no AC. It had an atwood gas furnace, a two burner stove, a small sink, and a portable porta-potty that was emptied by taking the bucket out and dumping it in a toilet somewhere. We had our two young boys and one of their friends with us. The van could sleep 4, two adults and two smaller people comfortably, but we mananged with 5. We traveled from North Carolina to Texas and back - in the summer time. We had a great time.
You say ask Suv or Class-B. Personally I'd take the Class B. If you take the SUV you will be stopping and staying in hotel/motels, most likely. If you take the Class-B you still have the opportunity if things should ot work out to your expectations of stopping at a hotel/motel. Now who would know when chekcing in that an Adult or maybe two adults are going to sleep in the Class B. Consider the scenario, and finances.
So I say go for it and enjoy the trip.
p.s., when we traveled with a 5th wheel we towed with a F350, King Ranch, Crew Cab, Long Bed. In 90% of the time if, there was maneuverability and a car could get into the space, so could I. I am guessing that it was as big as a Class-b for parking. I never had any problems at any historic sites that we visited. No, we didn't visit them all but it was not because we were concerned with parking.
Wayne MSGT USMC (Ret) & Earlene (CinCHouse)
2008 Winnebago Destination 39W
It is what it is, and then it is what you make of it.