I've been reading about your trip and I am wondering if the Ural is the right bike for you...70mph cruising? I am not sure the bike is really made for this! I have a 2011 GU and even if HE (for me it's a male!) feels OK at that kind of speed I trust the entire community saying HE is not done for that (specially loaded!). When on freeways I am trying to stay around 58/60mph (GPS) and at that speed it looks like able to go around the world...
As said before your problem could be anything... I'll start by the valves (more freeplay), ignition (a C1 or C2) and then carbs (too much oil in the air filter?).
I was reading with interest parts about your dogs...I have two Beagles and take them in the sidecar, with my wife and apart of the "cool" factor (as you pointed when no one showed interest...) I do not think it is a good idea! Even if my dogs, as any dog, love to sleep (and do it in the nose of the tub on their cushion), this is not the most comfy place for them! Adding that dogs are "routine" animals I am not sure that the extra stress of new smells, places, foods and continual movement is right for them!
As usual: this is only my opinion!!!!