I don't think your handlebar bag idea will work with a Corgi-the ones I've seen are pretty sturdy, if short and low to the ground, little dogs. I do think they would probably make good travelling partners, as you could carry them on the trailer, and let them off to run beside on uphill sections to lighten the load. They are herding dogs, so they can probably put some miles in on those little legs.
I've only met one person touring with a dog ( seen one or two others on the road also). He was touring from Winnipeg to Toronto with an 85 pound Akita, using a child trailer for the dog, and letting it walk uphill beside the bike. He didn't seem to have any complaints about travelling with the dog, and they had done over 1000 km together when we met them.