Swamped

You're going to integrate a harmonic oscillator into the workings of the vocalizer/amplifier, such that over time and repeated use the device will gradually produce more and more of a high-frequency audio pulse in the range that would interfere with the robots perceptions.

You wouldn't expect it to kick in for, oh, a few days... unless of course the robots start talking to each other, triggering additional oscillations. But hopefully it'll take a while, without any perceived disturbance until after you've been granted your freedom.

And regardless of whether they grant you your freedom, you may be able to use this act of sabotage (and the promise of a cure or the threat of imminent disruption) for leverage in the future.