Apologies Alistair, I obviously missed that part.
My (new) guess is the eccentric has worn a bit, so not enough pressure is being applied onto plate.
If when tightening the clamp, with the lever bar, and if it's moving up to the top of the movement it's probably not putting enough pressure on.
To check, after loading a plate, place a strip of thin plastic/packing into the clamp, then it may feel more solid when you tighten it.
Slight guess work from me, but I'd think you'd be able to remove the clamping mechanism without interfering with the register pins.
As an aside, I've never witnessed a press in a state like that. Washing up the rollers should not be causing that. Time for new wash-up blades perhaps...
Steve