Don't think you want to do that. When the immigration officer scans your passport it will come up on the system as being lost. And then you get to follow the kind agent into the back room.

Winterstale - if it was me I would not take the chance to get it screwed up. Since you are traveling to STT (airlines won't care) and BVI just needs a valid passport I would do the renewal after getting back from the trip. I don't see any upside to doing it before, i.e., as far as the BVI is concerned you have a valid passport now and a new one will be no more valid that the current one.