As far as I know, all of the 121 operators in this country do not allow their pilots to fly commercially anywhere else. The only exception is being a DPE since DPEs are technically paid for examining, not flying. The DPE is not supposed to touch controls and log the time for that matter. Simply said, flying occurs incidentally during the exam and only the applicant logs the time during the checkride as a PIC. I'm not sure what part 135 is like but I assume similar restrictions. Once you are working for 121 airline you can't fly and be paid for it anywhere on the side, instructing included. Sorry for the bed news!