It is possible to optimise the interface without knowing details of the remote control.
This requires some measurements to be made.
If there is only one switch then there is no multiplexing. This I have assumed.
Consider the terminals of the button A and B.
Measure the voltage at the terminals with respect to -ve of the battery using a digital multimeter. mark A and B on the circuit board.
Measure A and B without the button pressed.
Measure A and B with the button pressed.
There are quite a few options for an interface.
For example: If A is at the battery supply of 3.3v and B is 0v when the switch is open, then both are at 3.3V when the switch is closed.
In this case B needs to be pulled to 3.3V to operate the remote.
The safest way is to use 2 transistors a NPN followed by a PNP to switch 3.3V to the remote . This is called a "high side switch"
An example is the 3rd digram down on this page, but we would need lower value resistors:
It should also work by connecting the IO pin straight to B.
Putting a 100 ohm resistor between the IO pin and B would limit the current if a mistake had been made and would not usually prevent operation.
What is the supply current from the battery when it is transmitting?
What is the current through the switch when it is transmitting?
This is one case. If you state the voltages and currents measured, it should be possible to give more specific advise.