Agreed about not putting the probe inside the tub. Partly because the snake may move it etc etc, but also because you want to be able to easily move the tub around for various reasons. It's also going to need cleaning... there are plenty of reasons not to, especially for contact heat sources (ambient heat is sometimes an exception but you want to be very careful, and this is not such a case).
As I keep saying, the thermostat is not a thermometer. Ignore the reading on the thermostat. It can say 40, it can say 20, it's common for them not to be calibrated properly and often even if they are, the temperature setting won't reflect the temperature at the site of interest. If the *thermometer* says it's too cold, increase the setting on the thermostat, regardless of what that number is. If it's too hot, reduce the setting on the thermostat, regardless of what that number is. Set the temperature using a thermometer to measure temperature and using the thermostat setting as an arbitrary scale, or a vague reference. I've seen so many snakes kept at the wrong temperature because of thermostats being used as thermometers. Do not use the thermostat scale as a guage for setting the temperature. Use the temperature, measured by thermometers, to set your thermostat.