Welcome to FT.

I believe that ground services for KLM in the US are handled by DL, but I could be wrong.

In any case, I have traveled on multiple airlines with more than one ticket on numerous occasions. I have never had a problem checking luggage through to my final destination. I have always had my itineraries and e-ticket receipts for all flights in my possession. I have also written out the routing using the 3-letter airport codes and flight numbers. I have also gotten to the airport extra early since I know it involves manual entry of the downline flights in order to generate the luggage tag.

Delta, KLM and Air One all have ticketing/luggage/interline agreements in place. So theoretically you should be able to do it. I would be polite but persistent and be sure to get to the airport early. The worst case scenario is you trying to push your case with 100 people in line behind you wanting to get checked in.