Using lattice for tire deformation

Im using a lattice to deform a tire, which works perfectly for planar movement.
However, when moving vertically (from vehicle tilting when turning/accelerating) I want the tire to deform only on ground contact, not above it and not so it clips the surface plane.

I have been playing with the lattice deform strength setting - but it does not give me exactly what I want.

Maybe this image explains the issue some. I would like the plane to be a defined border for the tire mesh.

Animate the lattice itself?

That’s an idea!
Then the lattice can move up proportional to how the wheel center moves down. Any idea how to achieve this?

Nevermind. Animating the lattice worked!. By setting a driver that is (negatively) proportional to wheel mesh z-movement it sort of stops the tire where I want it to. At least I have enough parameters to tune this way better than before.