AltHold mode overshoots were known, since the copter was before tuning.
I went ahead and lowered PSC_ values to a level where the oscillation in loiter/poshold where not amplified, but kept in control. Then managed to do a successful autotune. Now in Althold the copter is locked in, and only PSC need some more tuning.
It seems the problem was the relatively small fuselage with long arms, strong motors, plus a payload extended below the copter, acting like a pendulum.
Learning; if the log says your PSC needs tuning then it needs tuning 