You definitely need a thermostat for any heat source, including UTH, do not run a UTH without a thermostat ... it will cause burns
Right now I'm using a Jump Start thermostat on my UTH like this:
https://www.amazon.com/Hydrofarm-MTP.../dp/B000NZZG3S
Heat mat is plugged into the thermostat, and the probe for the thermostat is either securely taped in between the mat and the tank or (like in my case, because I stuck the UTH on before I could do this), you could securely tape it to the non-sticky side of the UTH with aluminum tape ....but you want the probe monitoring the mat itself, and NOT placed inside the tank. If the stat probe is inside the tank it could be urinated on or shifted by the snake and then that could throw off the thermostat and cause possible overheating as well.
Second, you need a thermometer probe on top of the hot spot on the INSIDE of the tank to monitor and make sure that UTH keeps the inside surface that your BP touches at around 90 degrees
For my lamp, I use a lamp dimmer and manually adjust them depending on the weather or day vs night..... you could use thermostat as well for this for a more automated process (it would turn the light on/off to keep temp in check), but right now I'm just using the cheap dimmer and keep my eye on it. I'm in SoCal so our weather stays pretty moderate year round.
On top of all this, having an infrared temp gun is great for double checking any spot in the tank