Here's the backstory:
KVS showed 5 seats in business outbound; 4 seats on the return. Airfrance.us showed the same thing. DL's search engine showed the same thing but couldn't hold or book, giving the dreaded "One or more of the flights you selected just sold out" message. EF showed zero seats in O.

I called Delta to assist and spoke to an agent, her supervisor, the manual booking desk, the website support and a website support supervisor. They all were very sympathetic and tried to help me out, but no one could do anything. They tried to long sell the award, but it wouldn't work. I had a supervisor go as far as to call KLM twice to try to figure out why KLM was rejecting Delta's attempts to book the flights, but all he could figure out was that there were no seats in O.

Finally, I tried something a bit insane. I called KLM Flying Blue and asked them to put my desired itinerary on hold. Interestingly enough, the hold came back and listed "Class O" on klm.com. Recall that according to EF, there was zero availability in O. This was encouraging! I then called DL, got myself transferred to the manual booking desk, gave them my KL reservation reference code, and asked them to help me book the flight. Here's the crazy part: they booked it even though it was an itinerary that was created by a KL rep. Even more interestingly, I asked the DL rep to add a couple of connections on AlItalia (AMS->FCO, etc.), and she was able to do it. I looked at the result using KLM.com and could see the DL rep was just straight-up modifying the KL itinerary.

Given that Flying Blue is used for both KL and AF, I wouldn't be surprised if this same technique helps out with some of the AF / DL SM award space booking problems.

It's possible that everything I discovered was already known by the community, in which case I apologize for being unnecessarily sensationalist. I'm just excited because I finally managed to book my dream itinerary using what I think were some pretty clever tricks .