Sounds like a bad idea to me.

It will certainly complicate your trip, but more importantly, I can't see how it would be a good experience for your pet. Does it normally spend 100% of it's time in a cage or on a leash? Or does it run around in your house? What if it gets out of its carrier, is it going to come back soon or is it going to run away? WIll it sleep in your tent with you? Is it tent-trained? What about attracting other animals - bears, racoons... What about it's affect on wildlife, will it attack birds, rabbits, whatever? Will it try to hunt from your handlbars? WIll it jump out of it's basket, entangling itself in your wheels? WIll it get run over by a car?

Bad idea.