How to aerodynamics at speeds above Mach 3?

I’m just trying to tune the Sound Airplane as much as I can. So far I’m at Mach 5 above ground, however the elevator is becoming really unstable at speeds > M3.5.

Someone mentioned PID tuning, but I can only find PID settings in the propeller, which seems to be governor related.

What helped so far was to reduce the elevator effectiveness and max angle. But to me it also feels like two parts of the plane are flying into different directions and the elevator tries to compensate for that. For instance the tail wants to fly downwards, whereas the front wings create upwards lift

I did see some debug windows, but none of them seems to help me resolve my issues. Any ideas?