What is the possibility of an 88M getting stationed in Washington DC?

There are a few nominative assignments that you must compete and be selected for. You will need CSM mentorship at a minimum. See what is one ASK and call your branch manager. There is more than just the Pentagon. You have Ft Belvior, the Department of the Army's HHC and Garrison. Ft AP Hill. Be careful what you ask for, you might get it. I knew an 88M who was the SMA's driver.

Who in the hell would want to be stationed in that shit hole. I was once

Look at the White House Transportation office.