Interesting question, which I am throwing an answer at because it's well off the beaten track and could hang out there for awhile.
I don't believe there is a "congressional" golf course. Maybe there is!
There is a private Congressional Country Club in Bethesda, MD. I don't know if they offer courtesy access to midshipmen.
Midshipmen, as active duty, can play on any military base golf course, including the one at Naval Support Activity Annapolis. There are some military golf courses in amazing locations.
There is also the Army-Navy Country Club, a private club, with locations in Arlington and Fairfax, VA. Midshipmen can get a phenomenally low membership entry rate. Many USNA firsties take advantage of that before graduation, if they plan on being lifetime golfers, as there are local and remote annual dues. The initial membership fee climbs with rank.