So screws driven between the plies will not compromise plywood's
strength to any significant extent and will not separate the plies
from the weight, right?

you'll have to test that hypothesis on YOUR plywood with YOUR screws
and YOUR skins. test pretty agressively, too- bang it around some to
make sure.

me, I'd probably start with 1/4" crown x 1-1/2" staples, partly
because it's what I have and partly because the staples displace so
little wood.