Per AF lodging policy, not having pet-friendly TLFs doesn't make them inadequate...AFI 34-135, paragraph 1.19.6.
As far as space being "adequate," paragraph 3.3.3 says, "Lodging assigns families to [Civilian Lodging] on a voluntary basis when TLFs are fully occupied or adequate space is not available (lodging manager will determine if VQ space is suitable..."
So, that's the AFI for what it's worth. I'm not a fan that the lodging manager has the authority to determine the adequacy of their own space since their operation is funded by your per diem dollars...seems shady. However, convincing the lodging manager (or his/her boss, or boss's boss, or IG, etc) that the space is not adequate is up to you and your leadership.