tIn the case of soft bremsstrahlung, one only can observe the average energy loss in a large number of collisions (say, by measuring the rate at which a plasma cools by emitting bremstrahlung), which is not the same as observing the effects of radiation damping forces during the scattering process itself. The "unsolved" aspect of the problem in classical electrodynamics involving runaway solutions or acausal preacceleration does not show up in the low-frequency regime, and the effects of classical radiation damping are small, deflecting the electron trajectory by a miniscule amount compared to the Coulomb scattering event which gave rise to the bremsstrahlung in the first place. Thus, apart from its statistical effects on energy loss, "true" classical radiation damping is not observable in bremsstrahlung .
Another way of looking at it is that the Schott energy term in the radiation damping equation is a correction to the elecron mass-energy which presumably should impact the Coulomb scattering cross section since the latter depends on electron mass, but a simple back-of-the-envelope calulation shows that this effect is far too small to measure it in the lab, sadly. Thus, no experiment involving bremsstrahlung in the classical regime can distinguish between competing models of RR forces or confirm the existence of the Schott energy of the electron.
Even disregarding lack of experimental access and only considering QED vs. classical theory, it is far from clear (to me) at what QED order one could distinguish between competing classical expressions for the RR force (which both result in the same average energy loss and this energy loss is recreated in the soft photon limit of QED perturbation theory).
The classical expressions which differ in their handling of the Schott energy term but which have the same overall energy loss to lowest order are the ones we want to distinguish.
