Have you tried shooting without using the in-camera shutter to see if it really is the shutter that causes the problem? Shoot in a dark room with a long shutter speed and use room lights to control the exposure time.
No. I've never done any tests to try and separate mirror slap from shutter vibration. I read years ago that the critical shutter speeds when mirror slap can be a problem are from 2 secs to 1/60th. I've never found a problem outside this range. If the shutter itself also causes a degree of image-degrading vibration, I don't see what can be done about it if it's in the same shutter speed range where mirror slap is also a problem, except use a faster shutter speed.
I was surprised that I couldn't see any difference with the 20D in this range from 2 secs to 1/60th, comparing images with and without MLU enabled. The 5D behaved more typically with the greatest blur at 1/30th without MLU, slightly less at 1/15th and almost none at 1/60th.