The paper is just usually close enough, you can subtract the thickness of your paper, too.
If I want to be real close, I'll touch till the paper tears, or you can use a volt meter on ohms:
There is a lot of work holding ideas, just do a search here, like carpet tape, or 2 pcs. of tape and super glue, wax.
The problem you'll have is the aluminum plate will be warped a little, one side will bow out.
I like to use clamps bow side down, but you dont have room for that.
Hmmm, maybe a socket head screw at each end where the bows are (bow side up), where the head just reaches over the edge of the part?
Also not sure what machine your using