They make such a "repeater" unit as a commercial product now? That makes things convenient! I did a similar installation several years ago, and used a small "ice cube" relay wired to the existing bell, operating a standard "push button" type wireless doorbell transmitter with the button removed and wired to the relay contacts.
The wireless bells (at least the NuTone ones I have used) work very reliably, in my experience, but the downside is the need for occasional battery replacements. Homeowners aren't used to replacing batteries in doorbells anymore (ever found one that still used the old #6 dry cells?
), and I have received a few calls that were resolved by a simple battery replacement. The battery in the transmitter is somewhat unusual, being a lithium type often used in RF car alarm transmitters.