Personally, I don't have a problem reading missions in continua I don't know. The reason is that I extract humor from how the agents are reacting more than anything else. If you're worried about the details of the canon going over our heads, focus the humor on the agents. Use more physical and visual types of humor—the spit-takes, the headdesking, the mad scramble for Bleeprin, etc., and especially any weird manifestations of the bad writing. The mental images will provide plenty of humor.

~Neshomeh

To be fair, it can work well even if your reader doesn't know the canon at all. I've recently been reading through a lot of older PPC stories, including a few set in the Fire Emblem 'verse. Despite having no former knowledge of Fire Emblem prior to reading the stories I still enjoyed them and got probably half the jokes involved.

I think a lot of boarders here skip the missions set in fandoms they don't really know. Mainly, I guess, because they think they would not get the joke. Or they read them and accept that they won't get all the jokes. If something ends up as a charge, that ought to be enough information that something was stupid to do. Explaining the jokes is thus not needed, as the people who know canon don't need it and others don't expect it.

If you think explaining is needed, you could make one of your agents an absolute noob about the fandom they got kicked into. Then everything would have to be explained to them.

Not all boarders know LotR (blasphemy, I know).

As you mentioned, it's hard to explain a joke without killing it.

The best way to do it when you don't know who'll know the canon?

Have it explained beforehand. Or have mental shortcuts in your explanation. Or give someone the role of not knowing the canon so you can explain it in character without talking to the fourth wall.

Make comparisons where it works and helps illustrate the point of ridiculous. "Hawk Laguz being the king of Serenes Forest? That's like having Elrond suddenly be Lord of Moria!" Looksee, that gets the point across to us doubly. People familiar with both get a doubly absurd image to be amused at, and those familiar with only LotR get to boggle at the wrongness it'd translate to in LotR-terms and imagine what it must mean for the canon being covered.

There are plenty of ways, just takes a bit of practice and retooling.