Hey, that's a Clouds, right? I have an eye out for this plane, looks super sleek and spacious.
Here is a graph of relevant quantities, for the climb to altitude segment.
At about t=945, you command a full elevator pull-up and the NavPitch goes to 20 degrees, as it should. You stay there for 1.2s, and in that interval the actual pitch goes from -2 to +14 degrees. I admit that this would feel kind of sluggish, but I don't see any significant misbehaviour from the controller part.
If you want to make the pitch controller more aggressive, then I would increase the P-gain further.
In case of overshoot+oscillations I would increase the D-gain and in case of oscillations in steady-state, I would increase PTCH2SRV_TCONST.
I don't see any significant oscillations in the last log you sent. I'll check the next-to-last in a bit.