I connected the relay to the push button switched in your diagram, i powered the relay with the autogate backup battery, used the sonoff ewelink integration and exposed in to homekit as a gate which can be see in carplay when you reach near your home. If you are using home assistant, let me know, i will share my code to expose it as a gate. I also used an aqara door and window sensor to check if the gate is open or close. So in the home app, if i click once it will open both sides fully, but if i click again while it is opening it will stop, and then i click again to close. You cant open one side with this.

Another way you can do is, use a relay with the gate remote, you will need to solder.