I've considered this some more, and decided on the following changes, starting next week:

Monday (Volume):
3 x 5 Back Squats
3 x 5 Bench Press
3 x 5 Shoulder Press
Assistance: Weighted Chins

ETA: + 1x5 Deadlifts at a submax, but heavy weight; and + Upper Back work (as in Reverse DB Flys)


Wednesday (Light Day):
3 x 3 Front Squat
3 x 3 Push Press
Assistance: BB Curl

ETA: + Shoulder work (as in Lat Raises)

Friday (Intensity):
Cycle through 3RM - 2RM - 1RM
Deadlift
Bench Press
Shoulder Press
Assistance: BB Row

ETA: + 1x5 reps at a lower weight

This has the advantage of including Squat and regular Press, without leaving out heavy DLs, or Bench Press (which I enjoy). It ups the volume a bit on Mon, while likely reducing the weight a bvit (Squat vs DL).

The program as such works fine, no need to switch to something different.

ETA:  Changes to improve DLs, and climbing strength (upper back, which is where I'm lacking!)