Is there a fee I can pay to escort my child to the airline seating area and stay until her flight is airborne?

I've done this before and I don't recall having a fee. I would call the airline directly and see what you can do

No TSA regulations forbid anyone not boarding a plane goimg past security. You would have to pay a fee so that an employee of the airline will escort your child to the waiting area of the gate, and will wait with them there.

That is through the airline. They can do an unaccompanied minor ticket, but you will need to accompany your child to the boarding area and they will take them from there to the plane.