You could do it, but why would you? Usually, the BP swap is done if you're trying to make an earlier flight. Give someone already checked in a few bills to sit around the airport and take you're later flight, you take their earlier flight.

But, swapping out airlines, that may get a little dicey, since who knows what happens if the flight is canceled, diverted due to mechanical issues or something of that nature. In the OPs case, I wouldn't do it across airlines, although it wouldn't be because of TSA or anything.