Are there any languages common to the Irim Dwegsho-in-exile that would actually be used in the game?
Not really. You might occasionally run across some technical manuals or literature in Engi, Urtragian, Old Irit, or Galactic Standard, but those aren't exactly frequent outside the Dwegsho. Most everyone educated or who travels or deals with travellers knows Anshari Common (which evolved from the drow trade language in ages past). In short: No, it is very unlikely.
That said, this discussion on languages has got me thinking about giving everyone a free language corresponding to their place of origin, for flavor/worldbuilding purposes. Anshar's isolated valleys does produce a greater diversity of cultures than D&D's standard language system assumes. It's not something I expect or intend to come up, and I definitely don't want to play the translator game, but it's the sort of thing that would happen in a world like this.
