Considering the kind of goofball, gangbanger, political, protest, or other objectionable "events" that could occur at national monuments (with very real risk of damage to the monuments), if allowing dancing/singing sets a precedent, I think a universal ban on all 'events' at national monuments is a good idea. There's plenty of other places to exercise your First Amendment.
I am fairly certain a minute of silent prayer would not be in violation of these rules. But Joe would get busted for his excellent MLK impersonation. LOL
I was (peacefully) protesting at the Lincoln Memorial with a crowd of thousands in 1972 (NOT the Vietnam war), and people were sitting in Abe's lap, and I did not appreciate that at all.
Yet the WBC can exercise their first amendment rights @ soldiers funerals. Go figure....