If you put it on the CO2, then any residual sugar that ferments will just raise the pressure in the keg slowly above the set pressure. The keg will come back down to the regulator pressure every pour as you increase the headspace above the beer and equilibrate there. If you are pouring regularly, your rate of CO2 production in the keg from the sugar will be much less than the rate of pouring, and so it won't make any difference. The rate of sugar fermentation in the kegerator will be very slow anyway, as you're way below room temperature. Most likely it will stop, and the yeast will drop out anyway.
In the worst case, if you pour a few pints, then leave the keg for a while and it does overcarb a bit, you've only got to pull PRV and let it equilibrate over a day or so.
Biggest problem might be that the beer tastes a little sweet.