Lager fermentation in FRC445GB

So, I have a Frigidaire FRC445GB that will be converted to a kegerator soon.

In the meatime, I am planning on using it to ferment and lager a Pils. The fridge runs too cold for fermentation, and I would rather not drop the $75 on a new thermostat (i did just buy all of the hardware for a kegerator recently...), and would also rather not mess around with the swamp cooler thing.

Preparing for the conversion, I removed the factory thermostat from the side wall, and pushed it to the back (along with the copper tube temperature sensor). After doing this while brainstorming the best way to handle keeping a consistent ferment temp I had a thought:

since the fridge won't go warm enough, what happens if I just keep the copper temperature sensor in ice water? seems to me like that would trick the fridge into thinking it was cooler that it really is, and be easier to maintain 50ish degrees. There is no way that can damage the sensor, right?

thoughts?

I have no idea if the water will hurt the sensor, but if you put it in ice water, the fridge will think that the temp is 32*F inside and will not kick on at all. The thermostat won't power up the compressor until the ice has completely melted and the water warms up to 40*F or whatever. Then it will kick on until the water has cooled off to below 40*F.

If you keep changing out the ice-water, the fridge will never come on.

If you don't keep changing the ice-water, you'll end up with a fridge that is still maintaining 40*F, but it's basing that on the temp of the liquid and not the air, which (I think) will give you a chamber that is taking larger swings above and below 40*F than just having the sensor monitoring the air.