For easiness, you might take a comfortably large backpack and put the dog in it. I have one from GW Bass that is about 6"x9"x17". It has a drawstring and flap at the top. I used it when my JRT was injured and needed to go to the vet. I put it on backwards so the pouch was in front of me and loaded the dog in. He loved the idea that now he was getting the "big dog" view as we went down the street. He was able to perch his front paws on the top of the bag, but its softness made it hard for him to climb out. Admittedly, I was walking, but it might also work cycling.

The only problem was that when he saw the vet's office, he was not pleased.