I have a chinchilla and they are also sensitive to heat. I don't take him on my bike, but I did go to Lowe's and buy a 1' square of marble tile for him to lie on when it's too hot. I believe it cost $1.49 and it does a very good job of keeping him cool. Not sure how much heat it can deal with though. Maybe some type of cold pack in the bottom of your basket with the square of marble on top would do it.

I really wouldn't want to take an animal like that on a long tour though. Way to much stress for the both of you.