I am afraid that the final PID closed loop simulation on the web tool cannot be easily adjusted to add limits on the simulated PID.
You can take the identified model parameters from the PIDTuner and use them to create a custom simulation on Python or Matlab.
The same goes for the tuning provided by the tool, it gives some initial tuning with a simple slider for adjustment, if you wish to fix the overshoot, you have to de-tune the gains (slider to the left) until no overshoot is seen, or set your own PID tuning rules in an external Python or Matlab simulation environment.