Why complicate the simple?
Given that the bell is powered by just two D-cells, and it and the switch all work, re-wiring will be compatible, and the the easiest and cheapest option. They could carry it out themselves if reasonably practical: it doesn’t need an electrician.
With all other components still in place, it needs only low-voltage twin cable, e.g. loudspeaker or telephone-extension lead, and some way of fixing it in place.
Trying to convert the existing bell to wireless operation won’t be easy and likely cost a lot more than a reel of wire.
The age and design of the house are not really relevant. They didn’t have electric bells and dry-cells in the 1800s.
Strange that the original cable was “pinched”. Its scrap value would be tiny.