New doorbell stopped working, now not working properly
Apparently, the new doorbell is made to operate at 16 volts not 10. It seems to not work well under-voltaged.

I would install the new transformer and disconnect the wireless extender for a while until you determine if the new door bell is working properly. Just make sure you shut off the circuit breaker feeding the old transformer before replacing it. Double check to make sure power is shut off before replacing it.

Good luck!