Relative to over-controlling when the tail wheel lifts, if your plane rolls straight with the push test, then you may have a built-in imbalance between rudder authority at a given stick position and the tail wheel authority at the same stick position. This can be a frequent problem when the tail wheel is driven by a non-adjustable direct connection to the rudder rather than driven by the servo or an adjustable connection. This may explain why the problem lessens the faster you go with the wheels down. Typically a tail wheel in contact with the ground exerts a more positive yaw control than the rudder in the air at low speeds. If you go fast enough before the tail wheel lifts, then the rudder authority has caught up with or exceeded the tail wheel authority, hence there is no noticeable change in yaw when the tail wheel lifts.

If you can make your tail wheel throw adjustable, tame it down so that it is less responsive and more in line with the rudder authority at slow speed.