Have you considered the DL154/AZ7607 nonstop from JFK to VCE and then continuing from VCE to PMO/Palermo or CTA/Catania? It is DL metal to VCE but an AZ codeshare. I think you could still only use 1 award even though you will fly DL metal to VCE then AZ metal to PMO or CTA. However, I believe AZ flights from VCE to PMO and CTA have a stopover in NAP. No change of plane though.

Also, if you still want to use a DL SkyMiles award and DL and AZ have no availability in Business, look into using AF from JFK or EWR to FCO or MXP via CDG. Then use more DL miles for an AZ domestic award from FCO or MXP to PMO or CTA. Alternatively, consider the DL nonstop from JFK to NCE then use AZ from NCE to PMO or CTA via FCO. Obviously, this is not the most optimal way of getting to Sicily given the number of connections and the use of 2 awards. However, it can get you to where you want to go.

Unfortunately DL doesn't offer a SkyMiles award which allows for travel on multiple SkyMiles members.

As for AA miles, you can use swiss (LX) from JKF to Bologna (BLQ) via ZRH. Then use DL miles for AZ from BLQ to PMO. No AZ flights from BLQ to CTA.

Hope these suggestions help.