Given your comments I have to ask. Why not just skip the bike and walk? It seems like at 5-10 miles per day that would make more sense. Actually you ought to be able to make better mileage than that even walking. You could pack your gear in a baby jogger. I met a guy running across the country who did that and another walking across the country who used one. Your dogs could help pull or not. It would be easier to manage that than with a bike.

Finding places to stay every 5-10 miles seems unlikely anywhere I'd want to travel, but you may be able to find places to travel where that would work for short trips. Generally distance between services dictates the length of the day's mileage at least some of the time. Sometimes that is going to mean much longer days unless you are very selective in where you travel. I've never traveled anywhere that it would be convenient to do 5-10 mile days.

By the way where there are long hills the dogs will be pretty taxed going up and of no use going down. I have trail run and mountain biked with a young healthy dog and our paces were incompatible enough that it required a lot of patience. A rider is slower going up and a dog is much slower going down on the hills. Leaving the bike home and running made us much more compatible. It was kinder to the dog. She would turn herself inside out to stay with me on the downhills, but it would be cruel to expect it. I think she'd have run until she dropped dead if I asked her to when she was young. We have to use good judgement with our 4 legged friends because they will do whatever we ask.