Kay, so while I was doing the "For the Republic, Part II" quest, I told Moores that the Brotherhood wanted to ally with the NCR. I was something around Idolized before that, and once I told her that I became a "Good-Natured Rascal." Because of that, I cannot get the NCR Safehouse key from Hsu. Does anyone know how I can get my fame back up besides doing quests (considering I've done almost every single quest related to them in the game)?

You can give Legion ears to that one guy in Camp Forlorn Hope. He's in the Barracks.

"Good-Natured Rascal" is the top rep tier for it's "mixed rep" type. (wiki page w/rep 'fame' graph" - http://fallout.wikia.com/wiki/Liked )

You do not ever lose "bad rep"....once you gain enough neg. rep to be pushed into one of the "mixed" rep-tiers, you cannot go back. eg, if you want the NCR Safehouse, I believe you're going to have to reload. (or use console commands, if you're on PC)

The only Faction you can reset rep with is Freeside.