You have award tickets purchased with DL miles. This means that DL can open space for you (if DL is still selling four seats on the flights in your cabin class) on any DL operated flights that work, but DL cannot force partner award space.

Your choices are either to take one of the options you list or find some other way to fly to JFK on DL that arrives at an appropriate time for your AZ flight to FCO. You can also try to connect from the west coast to AMS/LHR/CDG and then connect to a SkyTeam partner flight to FCO, but this would require that award space be available on the intraEU connecting flight which might or might not be easy. Your chances might be better if the flight is offered as a DL codeshare.