My first inclination would be to put a better motor controller that would regulate first current, then speed. By this method when the bit is loaded, the controller would source more current to the drive to maintain a set velocity.
But you suggested perhaps doing something to the flywheel/pulley, and yes indeed there is a simpler and cheaper solution, and one based in fundamental physics rather than brute force engineering.
The one thing you will give up is that initially it will take a bit longer for the system to accelerate and come up to speed. But once it does it will better resist the torque loading applied when you split the wood.
So as to maintain the same speed, keep the larger pulley as is, but attach to that same shaft another flywheel that is larger diameter and mostly weighted around its circumference. What this will achieve is a higher moment of inertia for the upper shaft and the ability to store more angular momentum. It appears that when you load the splitting bit it isn't too long before the log is split, so the wheel just needs to be sized accordingly. Without knowing, even approximately what the speed and torque loading is, it's hard to say how much you would need to increase.
Better to keep larger diameter, lower weight than smaller diameter, larger weight so that you don't side load the bearings as much and wear them out.
