So you move every year or 2 and are considering investing in a bigger truck and a huge 5ver if you can find one big enough just to not stay in a hotel while on the road for a few nights a year tops and a few family camping trips?
First, if you have some "light duty" towing experience, a 30+' 5ver with a 3klb car wagging it's tail is a Whole different level of towing.
Second, you'd be spending $50-100k to save like $500 a year in hotel bills? Plus if you move that often, does your employer not pay for that stuff?
Third, if you are buying a truck and camper anyway and insist on towing the car with the same rig then think truck camper on top of that big 1 ton you'd need for the 5ver then you can tow dolly or flat tow the car behind with ease.
Last, I've moved every 2-1/2 years avg or less for the last 15 years and the last thing I want to do is set up a camper every night when I'm in moving mode. Except for the trip to AK last year, camped a few times on the way up for fun. Typically I want to shut the truck off, feed the family and dogs and catch a comfortable bed in a hotel. And while I always end up towing something substantial on the moving road trip, frankly it's a pain in the @ss on the longer trips.
I don't see where this plan is anything but a money pit and by your questions, you are planning on breaking even or saving $ by doing this.
IMO get a camper if you want to go camping, make things easy on yourself while you're moving.
Not trying to be a duck, but offering another perspective from someone who moves long distances often as well.