Just take a regular cheap RF garage door remote, put it in the fancier doorbell case if it will fit, and put the appropriate decoder and power supply in the house, perhaps using the existing chime decoder to get the ding dong sound. I would guess the doorbells are just an old design, not using the modern advances like SAW transmitter filters, receivers with similar SAW input filters and better designed decoders that do a much better job. All the RF doorbells i have looked at as old and busted invariably use a very outdated RF transmitter, with either a SRBP board with cheap hand wound coils on it, or slightly better FR4 board with similar cheap junky hand assembled coils and adjustments, and using the cheapest of components and no decoupling at all.
Modern remote receivers and transmitters are also available with better code algorithms, though it is not too bad to just have a single repeating code system instead of a rolling code, as you are not going to be overly concerned too much with false triggers providing they do not occur too much, and even there the decoders only respond after receiving 3 or more good data frames immediately after the other.