The AC should run just fine while traveling, assuming you have the generator power to run it. Perhaps, tho, you should try staying in the garage area while someone tows you around--just to see what it is like, not sure this is a good deal for pets...
Consider, tho, if you have a propane gen, and use it for the AC, it can go thru a 40lb bottle in about 24-30 hours, or quicker. Varies with gen size and AC load.
I myself, would not subject our dogs (Yorkies) to that type of travel.
Hope you find a solution--how about a motorhome? That would work for people and pets.
2016 Tiffin 40 QBH
2015 38RSSA, traded
2005 TK3 #1869, 10 yrs of memories,
2017 F450 KR--one more Ford is it
2009 F450 4x4-died; 2010 F450-retired