I'd recommend building separate continuity tester. I'm sure you can find a circuit online. Or by a meter with the feature built-in, or buy a separate continuity tester.
If you add circuitry to the meter you'll likely mess-up the other functions. I found the datasheet for the chip in your meter and it doesn't show a continuity/buzzer output.
I'm surprised it was fried by reverse voltage but "reversed" diodes across the LED & phototransistor will allow a current path to bypass the internal components when current flows in the opposite direction. (The existing resistors will protect against excess current through the protection diodes.)