He should go. Great comments on safety, etc. I'm envious that you have a small pup. I have seriously considered getting a hack so my 110 lbs. buddy can come with me/us. Kids are all grown/gone and he loves traveling, boating, camping, swimming in freezing mountain streams, running on the beach and playing in the ocean. He's a well behaved total clown that adds to our fun and is always a "welcome back" guest at hotels and friends/family homes on our travels.
Over 30 years ago I road in Enduro's where an older gent had his dog ride on his gas tank with nothing but a secured carpet remnant. They flew through the woods and clearly the dog LOVED it.
Fast forward to today. I live near the Blue Ridge Parkway and I routinely meet folks traveling cross country that have their little dogs riding in nylon or custom leather made accomodations including "Snoopy" goggles and helmets. Don't overlook Beemerchef's extended travels with his pooch.
This is a trip for you guys to make the most of, do you two want him along? With proper planning it will be a blast for the humans as well as the pup. They're more adaptable than we are and a lot tougher LOL.
Would the bears and big dogs still be out there if you traveled with him in a cage and stopped to hike, etc.?? I think so. Don't be careless, just be prudent.
Safe travels and congrats on having a GREAT SWMBO and pooch. All that and a motorcycle. You are richer than all the money that shitty job could provide. Enjoy every second this extra time off can provide.