Edit: I just realized the serial monitor only captures what is sent over the usb connection, not what is sent/received over the RX/TX pins. I suppose the only way to monitor these pins is to use another serial cable?
What is the difference? What is sent over the USB connection is what is on the Rx/Tx pins. As an example, in this project here:http://www.gammon.com.au/forum/?id=11283
I have connected a RFID card-reader to pin D0 (the Rx pin). I receive the data in the sketch, and echo it on D1 (the Tx pin). That shows up in the serial monitor, as I expect it to.
Is there a way I can echo the RX data without having to actually write it back to the serial port?
Well the serial monitor never echoes the Rx data, so that is the real problem.
For debugging you could disconnect the Tx pin from your bluetooth module (so the data from your sketch is not echoed back to it) and then add the line you had which copies the Rx data to the Tx pin.
Alternatively use the NewSoftSerial library and connect the bluetooth module to other pins (which is exactly what PaulS said). Or, for debugging echo via (say) I2C to a second Arduino and monitor what is happening with that. Or, get a logic analyzer.