Balancing in a bike game

I'm pretty new to Unity and I'm trying to modify the Car game tutorial from the Unity asset store into a bike game. I use a simple 3d model for the bike using two wheel colliders, one for the front wheel and one for the rear wheel. The bike also uses a rigidbody component. At first the bike just kept falling over which makes sense since there is only one wheel collider in width so there is really not much to balance on. I solved that by moving the center of gravity under the base of the bike. This makes the bike almost impossible to flip over which is what I want but I think the tilt of the bike while cornering is too extreme. If possible I would like the bike not to tilt at all while cornering, but still keep the physics.

I've tried to move the center of gravity even further down which kind of fixes this problem but the I get these oscillations instead that start a whole lot of weird movements in the bike. If possible I would like to lock the bike rotations on the local z-axis(the axis in the forward direction). If I use the "Freeze rotation" option on the rigidbody, it only applies to the global coordinate system, so it doesn't work when the bike isn't perfectly aligned with the z-axis.

I guess my questions would be, am I going about this the right way? Is there any way within the method I'm using to control the amount of tilt on the bike when turning? Is there a way, programmatically, to "override" the rigidbody physics and reset the local z-axis rotation for the bike every frame?