Pure speculation, but my guess is that it might work...
From an engineering standpoint, it might be the cheapest to take the existing remotes, add a receiver and make the button in the door handle, basically another remote. Push the button on the handle, your new remote senses this and sends out it's signal, just like you had pushed your remote.
Is that what they did? I have no idea... But just try programming it in and see if it works.