Alaska only allows one partner per award (KLM and AF count as one partner because it's the same loyalty program). So you can redeem Alaska miles to fly either KLM/AF or Delta. Not both.

There is a different award price for each partner even if flying to the same destination.

Note that you CAN book one award with Alaska miles on Delta to get across the ocean and then a separate award on KLM/AF within Europe. But they would be priced separately.

I think you only see BA awards because they are the only partner you can book the entire way. Try looking at the results you found on Delta's website and break up your search on Alaska's site. Can you book the same itinerary as two awards?