Another John brought next idea :
The problem with the battery-driven one is that the bell push is
extremely unreliable - some days it works, some days not. It's on the
porch which faces directly south; thus, it's exposed to the extremes of
weather. I've replaced the battery 3 or 4 times in its life (and the
battery holder seems pretty flimsy to me), and I have sealed the
bell-push with various gunges round the edges every now and again to try
and keep out damp -- but after a period of solid functionality, it
always becomes unreliable again.

So the problem is a poor quality bell push button?

Why not track down a better quality push button, which is waterproof
and connect it in parallel with the existing one?

I have two remote controls for my roller shutter garage door, one in
the house, other in the car. The buttons of which became iffy after a
few years and I could never lay my hand on them. I simply wired a
centre off, up / down switch to each and fixed the switches one in the
car, one in the house.