Bad Idea.
My horror story is that I was riding along with my sheltie at the side. She was pretty well trained, she had passed her obediance courses and was very close to me. I'd ride slowly and she would trot along. Everything went well until one day she noticed a pack of dogs behind her.
Then she tried to run away from them. Straight under my bike. The rear wheel went over her throat, and that dropped the bike.
Fortunately she was not hurt seriously. By the time I was finished crashing, she had picked herself up and was a block away and running for home at top speed. I got on the bike and pedaled as fast as I could but never caught sight of her till I arrived home.
Point is that even the best trained dog is still a dog. That lends an element of unpredictability to things. One that could get you both hurt.