Hi Walter,
Welcome to FT.

Couple of thoughts.

1st - DL will be switching from the 767-300 to the 767-400 for JFK-IST (DL72/73) starting on June 1st.

I'm not sure that all of the 764s (welcome to FT.. this is how we talk here ) are lie flat, but some are.

The bigger problem will be finding Biz class award availability in August, which as was pointed out above is high season.

You can get into the DL forum here, and/or seatguru.com to learn about the seating/services offered.

2nd - the gateway idea is not bad, but keep in mind the limitations.
If you buy the tix separately and there is a problem, the other airline won't really care or do much to help you should a delay on the other airline delay you.

also, there is a baggage problem. The low cost carriers will own your first-born if you try to bring TATL (transatlantic) biz-class type luggage onto the connecting flight.

Finally, keep in mind that while DL is inferior to some other options across the pond in biz, your connecting flight will be in intra-Europe-style biz, which means 2.5-4 hours in a narrow body with the middle seat blocked off. Hardly luxury.

Me? I'd do the DL if it was available.

Good luck,

R