I may be wrong, but I don't think so. In the past this would have been possible with paper passes but the problem today is that everything is electronic and all SKW (AND XJT/ASA) flight reservations/ticketing/boarding passes are done using a major partners own reservation system. Basically you need to an eligible nonrev in the system under which the flight is operated, and each major has their own rules about who they will allow.

DAL for example is not going to allow SKW add an XJT pilot who works in the UA/CAL system to the DAL nonrev system.

You should have JS priority ahead of all other carriers except SKW (You will be equal with ASA).