You may be able to stick a gate and/or some transistors in there (I can't think of a gate that will work off the top of my head, but that doesn't mean it can't be done), but if you're going to that much trouble you may as well go ahead and multiplex the I/O. Of course, your mux/demux circuit would have to be between the arduino and the BT module (and your other device), so it may not be all that feasible.
There are a few ways to get real hardware UARTs for extra serial ports, but I haven't gotten a chance to work through the (non-trivial) details yet.
-j