Uh, Arctic Birch is probably not possible as birch doesn't grow in the Arctic, perhaps you're thinking of Baltic Birch, a favorite of British speaker builders?
From the pictures on the web of the typical Mk. IV cabinet construction, its a thick walled plywood construction, with either dado or overlapping joints, plus nailed or screwed in battens (or fillets?) supports for the screw in front and back baffles. The batten also give some level of bracing. This type of construction is somewhat like typical BBC cabinets except BBC uses thin walls with bitumen pad wall damping. As such, it wouldn't be physically difficult to have someone cut the lumber, cut the joints, and then glue it up making sure its square all around. Once built and front and back baffles fitted and screwed in, the cabinet can be veneered to taste.
I would think if one asks around, or even looks for someone on CL with woodworking skills, you might find a hobbyist or home pro nearby who has woodworking experience, a nice table saw, and could do this relatively easily.
Looking at hertzdonut (Jeff) thread on his JBL L50 save, I bet he could easily cut panels to your measurements, cut the dado joints, and flat-pack ship it out to you.