Does the heat lamp give off heat?
If it does, it NEEDS a thermostat. Period.
Addionally, you haven’t addressed the mouse fuzzy issue yet.
Do you plan to get appropriate size prey?
Not tying to be difficult here. However. For the safety of your snake(s) all heating elements require a thermostat. It’s also unhealthy to underfeed a snake.
Please address both these issues ASAP.