well I suppose you could use a battery to power the normal doorbell
button with no light and trip a solid state relay, that would power
the transformer just to ring bell.

taken futher a solar panel could keep the battery charged.

or heck go solar completely with LED lights you might be able to have
the button light up

probably cost a few hundred bucks, to save a dollar or two a year.

put the solar panel somewhere it cant be stolen. they can be costly.

someone has too much time on their hands

cut out a decent candy bar a day, at a buck each and save 300 to 400
bucks a year