I don't know of a cheap place to get a 27" rim with a fixed hub. If you had shorter cranks, I would say just get a 700c for the rear, keep the 27" front. Those cranks are a bit long though, so if you run a 700c rear, you'll probably want a big tire (38mm or so if it will fit).

For what its worth, the bike does have the 3 hallmarks of a cheap frame. Stem shifters, suicide levers and stamped drop-outs. The record was the bottom of the line for the "racing" style bikes.