Whelp! I found the problem.

As luck would have it, I think I found not just one, but two ZX600A engines for sale near me. Same seller. I’m planning on going to go take a look at them on Tuesday. 

Is there any particular things I should be looking for? I figure if I can come to a good price with the seller, I want to get both motors, a radiator, and as many of the cooling hoses as I can find. If I can spot a set of carbs, I’ll grab those too. But should I look out for anything else? 

Im pretty sure the gpz600 oil cooler bolts up differently compared to your Zephyr 550 oil cooler so they won't interchange.

When I got my gpz550 I seriously considered doing this swap since there was an engine near me but decided to keep it "original". It looks like a lot of fun! I'll be following along.
 

Engine wiring and the ignition system if different from yours. What are your plans for the exhaust? You may want to grab the headers at least?

Upon further inspection of the sale listing, it looks like they have a whole at-least-mostly complete bike, plus the spare motor. There’s other loose parts I wouldn’t want, but it seems like there’s a good chance I’ll be able to get everything I’ll need. I’m getting kind of excited about this. It’s got me fantasizing about getting an AS Uotani SPII ignition for it. I’d make custom side decals that say “Zephyr 600R.”