After you saying by pressing the headstock against the wall, it made a difference, I thought of an old trick that a luthier you have all heard of taught me years ago....
One caveat, I have never tried this on a locking nut guitar, but it has worked a treat on non locking trems for me- I replaced the tuner buttons/knobs with a lighter material (ie. plastic or ebony). By changing the mass of the headstock, it might change the resonant frequency enough to alleviate or lessen the issue.
I believe the Suhr locking tuners are made by Schaller. Stew Mac has plenty of options for replacement buttons/knobs.
https://www.stewmac.com/Hardware_an...s/Tuner_Parts/Schaller_Replacement_Knobs.html
Another thing, and I'm sure you have done this already, but tighten the two screws that hold the locking nut in place.
Worth a shot.
Scotty A.