I managed to find the solution for the problem.
The lens calibration tool seems to work only with Live Link FIZ data present, despite there are other options that you can select neither of them seem to work. The following tutorial shows how to create a virtual Live Link subject to manually control focus, iris and zoom values if you don’t have an actual Live Link source from your physical camera: Camera Lens Calibration for Production | Unreal Engine 4.27 Documentation
With the virtual subject in place I can adjust the current focus, iris and zoom values used by lens calibration through the virtual Live Link subject.