In our area we can get a 3/4-inch thick solid core birch panel which is
much more stable (and more expensive) than plywood.  It has a birch layer
on both sides which finishes very well, and a solid material like very
fine particle board in-between.  It is a very fine material and can be
used for cabinetry.  I'm quite sure it is called something like solid
core birch paneling.  Check with a good lumber yard.

Re: nickel-plated slot-head screws.  Check with the antique automobile
restoration sources.  Nickel was commonly used until about 1930 even on
such as Model A Fords.  The biggest problem might be a somewhat limited
number of sizes available, although they did use both machine screws
and wood screws.  (Many car bodies were of wood framed construction.) 
In restoring Model A's it is not uncommon to substitute stainless steel
for nickel-plated fasteners as polished stainless has more of a yellowish
cast like nickel, than bluish like chrome.

I wish I could give you some addresses but I am stuck in Florida for the
winter and do not have my car restoration resources available.  But find
an old car nut and he will steer you to some of the restoration supply
houses for cars.  Hope this helps.

Bob Conant¶
Endicott, NY¶
(Temporarily hiding out in Ft. Myers, FL)