I have made thousands of QSO's with my KX-1 and never had any problem "zero beating" , but we all have different ears. If the monitor tone is in fact a square wave, then I would expect energy at 600 Hz, 1800 Hz, and even possibly higher frequencies. If this is the case, then something as simple as a capacitor across the audio chain somewhere which would attenuate 1800 Hz significantly, but have little effect on 600 Hz...just experiment. Might even be able to put it in the headphone cord. I think this would help your normal CW reception, since it would cut off some of the high frequency hiss (and QRM) without the need to crank down the crystal filter too much. You could build an external 600 Hz oscillator, and mix it with the headphone feed, and set the tone level to match typical signals. However many of my non musician friends, just cannot match pitches easily of even a pure sine wave, so I really suspect the little zero beat indicator would be the best solution. My K3S has a tuning indicator which seems to work just fine, but I usually just "know" when the signal is at 600 Hz, even without the side tone turned on. I built a somewhat similar circuit to the KX-1 tuning indicator with different parts, and it worked fine for another project in nearly any on the air conditions. Another idea, the KX-1 would probably benefit from a sharp external audio filter which you could switch in and out. If it was set to a very sharp setting (like less than 100 Hz), You could just tune the signal for maximum volume and probably be within 50 Hz, which is plenty close 99 % of the time. This filter, may be useful occasionally since we all know the shape of the crystal response is not ideal at all settings. Personally if I did not have good pitch matching ability ( most hams do not) then I would go for the LED indicator. You might even be able to use it externally if you do not want to wire it into your rig.
Enjoy your KX-1 Rick KL7CW