I'm not sure about using a BEC buzzer. The ESCs that come with the Gremlin kit don't have a buzzer. Some ESCs have a low-volatage buzzer but most ESCs only send signals to the motors to create the beeping sounds.
The buzzer pads on the bottom of the Femto F3 are for a
small 5V buzzer like these. Every time the red LED on the FC lights up, it sounds the buzzer. Since you see the flashing red light, you know you have programmed it correctly in the betaflight modes. You just don't have a buzzer hooked up to those pads.
I bought a 5-pack of buzzers and put one on another quad running betaflight and set it up on AUX2. Works like a charm and I just hot glued it to the base plate between the arms. I haven't installed it on my gremlin yet, but you are right, there is not a whole lot of real estate left on the TJ frame even though they are the size of a toothpaste cap.