I want to tap the output of the chip directly, triggering the relay through a MOSFET to avoid overloading the IC.
Depends what exactly the chip outputs, it may not be possible to tap it directly.
The chip may be a clone of the common LM8560 (as it was extremely popular and many clones under various names were made over the nearly half century since it was introduced), but also may be something completely different.
So we really need the circuit around the alarm output/control.
Also take into consideration, most clock radios do not have any real buzzer, but use the audio amplifier section of the radio for that function, that is why I asked also for the part controlling the power for the radio.
In
an example of a very common circuit used with the LM8560 we would be looking for the D14, R16(?), C22 and Q3 - these are responsible for switching power to the radio. And also here the "buzzer" function goes via the radio chip by injecting the AL_OUT signal into the power amplifier input via C37 and R29.
Something similar I would also expect in your device, so you need to redraw the related circuitry...
