Because you are active duty and on PCS orders, you will have priority at any of the bases along the way, for their Lodges. The ones I see on your route:
ABQ - Kirtland AFB - AFL is 505-846-9653, www.kirtland.af.mil
OKC - Tinker AFB - AFL is 405-734-5095 or 2822, www.tinker.af.mil
ARK - Little Rock AFB - AFL is 501-987-6753, www.littlerock.af.mil
Sorry, nothing in the Nashville or Roanoke areas that have a lodge of any sort.
If unable to secure lodging at the bases (we have never had any luck with that, but hubby is retired USN), you might ask if a given hotel/lodging gives a military discount to active duty. MANY do.
Military lodges may not have any problem with the dog. Pet-friendly chains include Motel 6 and La Quinta, but other places have individual policies.
Donna