Okay more updates:
I wired the pickup back in as it was stock, and the same issue is present with the selector in the middle position. I also did some more analysis work on the stock and new pickups and here is what I was able to determine (following the guide at the 4:23 mark of this video:
(156) How to wire a series/parallel switch for humbucker pickups (guitar hack #2) - YouTube):
Stock Pickup (Unable to find any diagram that corresponds to this configuration):
North Coil Start: Green
North Coil Finish: Black
South Coil Start: White
South Coil Finish: Red
New Pickup (Correlates to Seymour Duncan):
North Coil Start: Green
North Coil Finish: Red
South Coil Start: Black
South Coil Finish: White
I also found another diagnostic step that maybe somebody can help me understand...It involves measuring resistance when one lead of the multimeter is connected to the bare/ground wire of the pickup:
On the stock pickup, the multimeter registered resistance readings as follows:
Black: 2.35 kOhms
Green: 0.0 Ohms
Red: ~0.9 MOhms
White: ~0.9 MOhms
On the new pickup, I performed the same test and got the "open circuit" or "no reading" result for all wires...nothing registered on the multimeter, just as if neither lead was connected to anything. The video that I got this test from used the terminology "hard-wired to earth (ground)" but I'm not sure how that would affect things.
SBMM customer support got back to me and said that it could be the rotary switch but I'm not sure how to test that, so I sent them a photo of the stock wiring before I de-soldered the stock pickup and am waiting to hear back.