How should I insulate a room built into a metal barn?
The links in the other answers are very informative!

In answer to your question:


  So: is the fiberglass a bad idea?


The answer is yes, at least in the exterior walls.

In answer to your question:


  If so, what should I do instead?


You have a bunch of options, none as cheap as simple fiberglass batts.  My personal recommendation would be rigid foam on the exterior walls and ceiling with fiberglass batts on the newly built walls.  Pricey but effective.

Essentially, you're going to be building a little tiny insulated house inside the barn without the need to weatherproof it.  At the risk of losing a few square feet of floor space, why not just build the room free-standing in the corner, with maybe a couple of framing attachments (e.g. longer ceiling joists) to the barn walls?  That way you can do a simple wall with sheetrock both sides and fiberglass insulation in all four walls and the ceiling.