Hi hokiestar. I am currently in the higher ed program at HGSE, and I can tell you a little about how it works for Harvard employees.
In our cohort, all of our part time people are also employees at Harvard, and they can use TAP to pay for courses. There are about five of them in a cohort of 40. I don't know if it helps or hurts to be an employee, but it certainly is possible!
There are two things that I highly recommend if you are interested in the program. First, take some courses through TAP before you apply in order to get to know professors and the program. Second, talk to Judy, the Director of the program. A few Harvard employees interested in the program sat in on some of our courses and talked to current students, and they arranged this through Judy.
Best of luck!