I'm a civilian corporate pilot with 5000 total and 1700 TPIC. I have prior 135 experience but nothing 121. I don't have any checkride bust or other bad boxes checked on my application, but the only good boxes I can check are Univiersity flight instructor, Part 141 instructor, Part 61 instructor, Tail wheel endorsement, and a Master's degree with 4.0s. I don't have a PE.

I could stay at my current job where the only things I can really do to improve my app would be more flight time, and the possibility of becoming a Safety Officer; Or, I have an opportunity to move to a new single airplane flight department where I would be the chief pilot/aviation department head. The new job is something I would not take except that it might help my application at FedEx. It entails moving my family somewhere I don't want to move, working for a company I don't really like, and it's much less stable (being a new flight department the boss might decide after a year that he doesn't like the expense of owning an airplane and I could be on the street).

I suppose my question is this: Do I have a good shot at getting an interview with my current app (plus more flight time and a Safety Officer position), or do I need to consider taking the other (cruddier) job?

What’s the gross weight of your current aircraft? Over 12,500?

Yes, 30,000 lbs. I log about 350 hrs a year and about 300 of that is as captain. The new job would also be 12,500 +.

Do you have 1,000 TPIC in an AC that has a GTOW greater than 12.5k that was operated under a 121 or 135 certificate? I see you say you have 135 time but how much and what type? The app is specific about what mins are and what is also preferred. I know of no one on property without 1,000 pic either military,121 or 135 in a AC with a GTOW 12.5k or greater.

None of the above are worth an extra square on the resume, IMO.
Get the safety job, get well above the mins on PIC and update regularly. With all that instructing, no one beat you to the job that you could hit up for a recommendation or PE?

Yes, all 1700 TPIC is in aircraft 12.5K or greater and about a third of it was from flying Part 135.

Zero point zero.

I don't see anything about connections in which is a monumental obstacle. I know many management people with sons and daughters in aviation universities with connections up the yang yang.

connections up the yang yang? I know where my yin yang is, but where is my yang yang?

you don't NEED connections, lots of guys hired without them. That said, additional duties (office jobs) go a long way.