Lord of the Rings - Third Age
I have a heavily-modded version of Rolemaster that heavily draws on Middle-Earth lore. It even specifies what character options (like race and class) are available in each Age. So you could easily have a Second- or Fourth-Age game, and your options would be accurate to the lore.
RM isn't exactly widespread, so I'd probably have to spend some time making pregenerated characters, and would have to teach the game mechanics as you go. Wouldn't be the first time for me.