Motor cutting out when using HC-05 Bluetooth module
Previously I was powering everything off of the 5V line on the Arduino, but I just switched to using an external battery pack of 4 AAs and that seems to work fine.


That suggests, that your power source was not strong enough. You didn't say, how you are powering the Arduino, but I guess you are doing so via USB. The USB port of a PC can only give you 500mA of current (at least without any extra configuration to get more). Also the parts on the Arduino (for example the diode between USB power and the 5V pin) can only give you a limited amount of current. Especially when the motor is starting, the needed current is rather big. That might have caused a short voltage dip, deep enough to irritate the HC-05 module.

Using an extra battery as power source for the motor eliminates that problem. Though I think you should be able to connect all parts to the same power source, as long as it is powerfull enough to power all parts (plus some overhead). In that case be sure to connect the power for the L293D and Arduino in parallel, so that the motor current does not have to flow through the Arduino (which could kill the voltage regulator or the diode).

Depending on your motor, it also might generate a huge amount of EMI (Electromagnetic Interference) either though air transmission or through the power lines, which might irritate the HC-05 enough. In that case the easiest option would be to add a second power source. Though I think the problem is with the power source as described above.