Big Lew,
I have no experience to offer you, unfortunately.

I have seen 1 guy touring with a (very old) shepherd, sitting proudly in his trailer. Judging by the amount of gear on the bike, this guy was camping too. This was in Europe, along the Danube. (Campsites in Europe are very dog friendly)

Here in Holland you see lots of people on day rides with their dogs, usually small. One couple I met a few months ago had 5 dogs with them (2 in a trailer, 2 in front baskets and one in a rear rack mounted box. Electric bikes made it easier.

It is something I would love to do in the future, so please, come back and give an update on your research!