Your self induced steering oscillation is probably an aligning torque camber driven phenomenon. Since caster is a steer-camber coupling mechanism, the Fx force driver can cause this high 'Q' vibration.

Just because your scrub radius is identified geometrically at some layout position, it doesn't mean that its 40mm under actual operating conditions. There is a load, pressure, camber and rim width sensitive Mx contribution to this dimension, too. I believe I posted some sample Mx/Fz function plots on the TTC Forum. If not, I can do so.
The reality is that all the TTC data is nice an tidy in a 2 axis plot but not actually relevant to a rolling tire on a flexible (and compliant) multi-bar suspension platform. You need to incorporate it into a dynamic or quasi-static model where the multi-surface tire characteristics bloom. Sure, in most all cases superposition works, but the Fz, alpha and gamma drivers need to be there. I'm pretty sure I posted such a 4D tire surface Matlab spline Simulink model of a tire test machine on the TTC site. Feed that with some open loop inputs from a simulated static braking test and comment on what you find. There could be +- 50 mm of dynamic S.R. variation and you should know by now what that means to your '40 mm scrub radius' label.

That's my positive feedback for now....