In past houses I have used this sort of wireless extension for a wired doorbell .

Basically the same electric pulse that rings the wired ringer also powered the transmitter to issue a radio pulse, which the remote wireless chime them receives and makes its own ‘doorbell’ sound.

I forget which brand made it at the time, but I do not recall having any problems with it working with the remote chime located in the basement.

In another house (with a basement office) I just intercepted the wired bell wiring and added a second wired chime in the basement. It was not 100% reliable until I increased the voltage/wattage of the feed transformer. Then it was plenty loud and consistent.