I went ahead and used large binder clips.

I pushed the two cabinets together, removed the clear windows adjacent to each other, wrapped the wood in that thick and pliable clear plastic shelf liner, then used binder clips every 3-4” across the inside of the whole frame of the windows, marrying the two cabinets together with a plastic seal. Worked like a charm and much less intensive.

Now, I did have to buy a jig saw and shave 1/4” off the inside edges of both top display doors. They wouldn’t shut otherwise.