I wouldn't put a ban on it either. The beauty of RM is that you can pretty much do whatever you want.... at a cost. In this case, there are going to be hefty penalties for what her wants to do. A 2H Battle Axe on a mount is a far cry from a Lance which was intended to be used on horseback.
First, will the rider be using the axe 1H or 2H? 1H allows him a better grip on the reigns or pommel to hold on, but the attack is weaker. 2H is a better attack, but holding staying on the horse is a lot harder.
Second, as PeterR mentions, how much does the rider want to split the roll for riding and attacking?
Third, what's the player's intent with the type of attack? Some battle axes have a spike on the tip which could allow the axe to be used as a makeshift lance and I would give lower penalties. Is he doing a charging attack or just attacking while sitting on a horse. That too is less penalizing as he's really not riding.
After the attack is made, I would have him roll Riding on the MM Table with an appropriate difficulty to see if he stays mounted after the attack. (Then throw in a +10 Fall/Crush attack on the rider if he is unhorsed.) There is a bonus to the attack for mounted weapons, but weigh it against the penalty for the type weapon he is using.
This is something we've always held players to in our campaigns; the type of horse. RM2 has Lesser and Greater Warhorses. These were specially trained for battle, used to the din and chaos of battle, trained to go charging into battle. The typical nag, riding horse, or farm horse is not meant to do that so trying to spur one into a charge at a weapon wielding opponent may not go so well.
If our players had a Warhorse, there was no issue doing a charge attack. If it was any other type of horse, the player needed a Riding Skill or (I forgot the other skill, darn it) or at the very least, an Empathy roll to see if he could coax the mount into a charging attack. Maybe that's nit-picky on our part, but there are horses specifically trained for that maneuver, and if that is the case, then there are horses that can't/don't do that maneuver as well. Players paid extra for the Warhorse so we made their purchase mean something.