I found a solution. Instead of setting the tristate bit of INT4 to be an input (i.e., bit 15 of TRISA = 1), I set it to be an output (TRISAbit15=0). Now, this doesn't make sense because as I've already stated, I'm using a mechanical switch to send a pulse to the INT4 pin when pressed... i.e., an input signal to the MCU. I have the INT4 pin connected to a 10K pullup resistor for the switch to work.
So, setting the pin/tristate bit as an output no longer causes the INT4 interrupt to trigger. The signal looks noisier than it should, as if it's "fighting" with another signal, but it works fine now. Still confused as to how this could work with the pin set as an output when it's receiving an input...