Help needed wiring 3 volt garage door opener

It is quite possible that the remote requires permanent power (from the battery it originally had installed). The push switch will be momentary, discharging the stored charge.
Could you try this in smaller stages, i.e. power the remote from the bike via your regulator? this would then confirm the remote.
next step, power it from the light circuit (still manually pressing the button.
Finally, it may be possible to fit a "second" momentary switch on the bike in parrallel with the one on the remote

 

The positive goes to the ring that holds the battery down.
The negative goes to a connector that is mounted on the board in the center of the ring.

For simplicity I would find a 2.7 volt zener diode and solder the cathode to the wring that holds the battery down and the anode to the connector that is mounted on the board. Then I would find and appropriate size resistor, starting with maybe 470 ohms, connect it in series with the zener. The resistor is what connects to the 12 volt source and the other to the return making sure that the polarity is correct.

I would thing that if you make a mistake and hook the battery up backwards your remoter will become toast.
And if too much voltage gets on the remoter it will become hosed.

Be careful.