If you go down this road, you must be 100% willing to take the offer.
However, I believe it is straightforward: I like this job, and I like working for you, but I have an offer to go fly XXX airplane for XXX money. Because I like this job and like working for you, I wanted to give you an opportunity to beat their offer (note: not match).
That isn't a "threat to leave" per se, but it could be construed as such because you have made it clear you have an offer to leave and any reasonable person will understand that you asking him to beat your offer is tantamount to "If you don't, I'm out."
Because that's exactly what it is and should be.
As for a raise for being PIC...PIC has legal responsibility per FARs and should be compensated for that responsibility. If your boss doesn't understand that - in this pilot hiring environment of all things - you should vote with your feet.