Hammering a sheet on a hard surface will thin the sheet and increase its size; probably not in a completely uniform way. You might be better off trimming the copper once it is in place. I'm not sure how many rivet holes you plan to drill, but drilling 3/8" stainless might not be much fun unless you are using a drill press, lubrication and the right drills.

Expect the beaten material to grow in size and of course, work harden if requiring deep draws. Leaving a margin is prudent lacking experience.
I'm with Blindstar on drilling stainless, use cobalt alloy screw machine drills to avoid certain agg, unless the SS is already piloted and you're back drilling.

Use a decent drilling lube and you should not have any significant problem with stainless.

Drill dry and be prepared to use a lot of drill bits and be frustrated.

Thanks for your responses. In my research I had found that the copper would draw up and harrden when textured. I will be having all forms cut by water jet from a computer disk. The holes in the stainless will also be cut by water jet. The holes will be drilled in the copper to matchthe stainless holes in the adhering process. I was just trying to decide if I needed to cut the copper forms smaller: now I am thinking a litlle larger than the stainless...or exact and expect a reveal edge of stainless.

Eleanor, perhaps I misunderstand your question, but I don't see why the material, be it copper, steel, or any plastic material, won't grow larger when thinned.

I don't work with copper very much at all, but all the steel and iron I've hammered at the forge take up more area when thinned. Rods become longer when hammered. A piece of 1/2" square steel 1 foot long can be flattened to a piece 1/4" x 1" by 1 ft., or 1/4" x 1/2" x 2 ft., or 1/4" x1/4" x 2 ft., and so on. The volumetric measure of the steel will not change (except you have to make allowance for scaling in hot work.)

However, if you wrinkle, pucker, fold, pleat, or raise bumps in the material without thinning it then yes, you will need more material. I assumed you will be hammering your copper against a rigid anvil, but perhaps you mean to use a relatively soft backing that will let you create relief in the metal without thinning it.

Good luck,

Mark