1) Can you identify the ground of the 5V radio. If yes, can you connect the ground of the 5V radio to your arduino ground ? Note: By connecting these two, you may introduce some ground loops that may affect reception / cause interference
2) Check if one side of the keypad is internally connected to the ground of the 5V radio.
3) If yes, Check if the GPIO of the arduino is 5V tolerant (some are). If it is, try to configure the GPIO output pin to be "Open-Drain" and connect the GPIO output of the arduino to one side of the tactile switch. The other side connect to your arduino ground. Polarity is important
4) This should work.