After re-reading this thread instead of bits and pieces, It seems on the surface to be servo tuning, we have seen it happen a lot, machines are sent incorrectly tuned, machine was sent out poorly or non-tuned.
But..., ball bar looks ok, does that negate servo tuning issue? maybe, but if it does what is left? the spindle.
It cuts good at sub 2k, regardless of tooling or cut params.
Then question, have you tried the same cuts, with same params, but using a conventional cut instead of climb cut? if this was a belt issue this might change it.
Did you remove the cover from the head and observe the belt with a flashlight to see if you see the belt wave as you increase rpm above 2k, usually if it gets a wave or harmonic in it you will see it widen in its path.
Have you watched the belt during a cut.
I would say have Haas tech check belt tension but he will probably not have a tension meter, and will have no idea what tension it needs to be anyway.
Couple things that did make a difference for me, recheck the machine level, but also make sure none of the feet are floating, objects like to balance on 3 points you may have a 4th floating.
On finish passes on non-linear geometry use G187
G187 Video Link
and last on a brand new machine, this is the only things I could see causing this, and it did come incorrectly programmed.