Get a three-wheel bike with the two wheels and a basket in front (which will also discourage people from calling it a tricycle ;)) and put your pet carrier in the basket, or just the pet if it has good instincts to stay close to you. They're sort of expensive, you can get a three wheeler with a bicycle on the back for under $1k but if you want the steering wheel at the back then look for '3 wheel dutch bicycle' - it's a design that's very popular in the Netherlands and there's one US company that either imports or manufactures them, albeit at considerable cost.