Whatever you decide, remember to add plenty of ventilation and a bit of weather protection, so the your pet doesn't get too hot and the bedding doesn't get wet. Also, build it so that the pooch can't jump out unless you want him to; and, build it big enough so that the pet(s) can stand up and turn around.
A miniature horse trailer would be really cute; but, if you could make the trailer so that the sides unlatch and fold down, you could attach or drape canvas over top and all of you could sleep in it, too. A 3' wide and 4' long floor with 3' foot high walls could fold down to make a 4x9 foot bed platform when you get to your destination.

...That's just a rough idea to play with....