...That said, I'll probably want to exit out of public sector law eventually and cash in on my experience at some point by going to the private sector (now think about the lawyers that work for Verizon whose job it is to influence/litigate FCC regulations). So T14ness may matter in that situation...
It's very interesting you say that. The current general counsel of Verizon, William Barr, is a GW alum. He was the main speaker at the GW presidential merit scholar's dinner the Thursday before the GW preview weekend at the end of March. I would say one of the main themes of his speech was, do what you want to do and what feels right to you, and don't worry about trying to follow the crowd.
I agree with this: do whatever you think makes the most sense for you and what you want to do. This is doubly true since you're not coming straight out of UG and you have your own thing going on.
My situation is different from yours but not completely different -- I was offered a full ride to GW and a hefty scholarship to U Va. I've been out of UG for a while and I live pretty close to DC. I went with GW because that was what made the most sense for my personal situation and for what I want to do as a lawyer.
Best of luck on your decision.