I suggested a temp gun because usually you can pick up a decent one for the same price as a digital thermometer with probe.
As for the thermostat. The urine would cause a liquid state over the thermostat seeing the probes are sealed this is not a proble. However it will create a denser mass creating a larger heat sink making it more accurite.
Having the probe under gives you an inaccurite reading it is more accurite having it where the snake would lay.