There's absolutely no reason not to allow it. First of all it's awesome and it's about having fun, and second of all there's no reason that through physics you couldn't do it.

We've done all manner of similar things in our games. And based upon the rules, it's entirely legit. Here's how it would be handled in 3.5 and Pathfinder (I don't play 4th Edition as it is terrible):

If you chose to attack axe-first, it would be treated as a charge attack for purposes of attacking. You and the beast would both take damage from the impact, based upon the damage tables for falling damage and falling objects. At that point I would make it a Strength check to hold onto the handle, likely at DC 20. For further rounds, I would require a Ride or Balance (Acrobatics) check of DC 20 to stay on the creature's back. It is huge, after all, and bucking around once it realizes you're there.

If you chose to not attack but simply jump onto the creature, again you would both take applicable impact damage. At that point I would have you make a DC 15 Ride or Acrocatics check per round since you have all hands open and usable. From that point you could also make Climb checks to get closer to the creature's head if need be.

The key here that your DM needs to learn is that the rules exist to make the story happen. The story does not exist so that rules can be employed.