Two metal rods vertically inside the bottle. Both full insulated except one is exposed at the top(in air), the other exposed at the bottom( submerged). When resistance of bottom exposed rod is equal to the resistance of top exposed rod then the bottom is now exposed to air, therefore empty. Pretty sure this is along the same lines as other suggestions but this is how I would do it. But the device would also go off if you filled the bottle too much and submerged the top exposed rod.
Also, +1 for the timer. You don't want this going off at midnight
You will form a electrolysis cell, and likely strip metal ions from the electrodes, potentially poisoning your bunnies. If you really must use electrodes, then only apply a small current on an infrequent basis (perhaps 0.1 seconds every hour, etc...) to make your measurement.
Depending on your level of expertise and/or motivation, and the assumption that you are looking at this as a challenge, here are some other ideas:
1. Optical, using the change in refractive index between air/water to alter the light path.
2. TDR - time domain reflectometry, wind a tight coil of wire on a hollow-former. Send down a fast pulses and measure the time they take to return (easier than it sounds, and you can average it over perhaps 1E6 pulses).
3. Wind a coil round the outside of the bottle, use this in a colpitts oscillator and measure the frequency change.
4. Put a colored ball in the bottle, use a webcam with something like the roborealm/opencv machine vision.