First, Thanks for your service, I'm retired Army and my Wife also served.
Second, our first RV was a 27 foot Keystone Tailgator, it was heavy and I towed it with both a 1500 and 2500 Avalanche both towed it fine.
The idea of using it for PCS moves is great, We did that for a few years and you can also claim any weight difference as part of a DITY move. also the JTR (joint travel regulation) has provisions for RV life while PCSing or on official travel.
The Tailgator was in the 7-9k weight range and towed fine across america for us. you may need airbags to keep the truck level, get the biggest/heaviest weight distro. hitch you can find, 10 or 12k, it's better to have heavier than you need and underwork it vs. over working an 8k hitch.
Good luck and PM me if you have any other questions about anything.
Gregg