The back wheel is skidding because one wheel no longer has enough contact with the ground Norm ,so if you can put enough height of any material on the side that is loosing contact with the ground it would fix the problem.
Looking at images of the ride on mower I can see what the problem is.
The pivoting front axle doesn't have enough movent ,it's bottoming out on one side when this happens the back wheels will no longer sit flat on the ground.
As we know.
A pivoting front axle on a ride-on mower is designed to enhance stability and traction when navigating uneven surfaces. Here's how it helps maintain contact with both rear wheels while operating on such terrains:
Articulation: The pivoting front axle allows the front wheels to articulate independently. This means that when one wheel encounters an obstacle or a dip, it can move up or down without affecting the other wheel directly. This articulation ensures that both rear wheels can maintain contact with the ground, as they are not forced to tilt or shift excessively due to the movement of the front wheels.
Weight Distribution: As the front axle pivots to follow the contours of the ground, the weight distribution over the rear wheels remains relatively balanced. This helps ensure that both rear wheels can grip the ground effectively, especially while cornering or traversing uneven terrain.
All that needs to be done is to change the design of the pivoting front axle so it has more movement ,that way both back wheels will be on the ground on uneven surfaces.
Personally if I went to the quad bike brakes I would use drum brakes as it doesn't look like too much work and I see these Quad bikes thrown out a lot.
I would modify the front axle for more movement as I think that would be the least amount of work and it would still look original.There looks like there is very little room for movement with the standard front axle.