I am only voicing my thoughts on this and I am by no means an authority on fabrication or camera building. 4mm sounds like a lot. I cannot think of a method to accurately measure the space inside a normally functioning camera.
One idea I had a while back would be to scavenge the pressure plate out of an old folding camera like an Ansco speedex or something like that (cheap, not sought after folding camera that used 120 film)
I think some combination of felt/velvet should get you through a testing phase at least and may work long-term.