Use existing wired doorbell to trigger Ring (or vice versa)

Hello, I'm considering purchasing a Ring Elite to use with PoE. I would actually like to position the camera at the electric gates to our house. We have an existing wired door bell / intercom which incorporates a keypad to open the gate (or it opens with a remote control or by pressing an intercom button inside the house). I would like a Ring to be able to answer the gate when I'm in the garden or away from the house altogether. However, I would like to leave the existing door entry which has its own bell push as well as having the Ring. Rather than having a sign saying "push the Ring bell" I'd like visitors to be able to push either bell and the chime will ring in the house AND Ring will send a message (I assume I could leave the old one connected and it would sound in the house, but that kind of defeats having Ring). I know that is possible if I just use the Ring bell push (and connect the existing chime), but I'm not sure if it would be possible to have the existing bell push also trigger the Ring? Does anyone know?! Sadly I can't remove the existing intercom / keypad (as for opening the gate remotely I will use a separate product for that - I think a solution exists in the US, but not in the UK yet - in any event digiGate looks like a good solution for that part). Thanks!, Nick