Yes, but only by a few degrees. As
JS stated, you will only be turning over the free turbine. It will move some air, but not enough. Your starter is the best option. And taking your last point, on shutdown the ITT will be cooled by passing air. As soon as the air stops, the residual heat in the engine dissipates into the surrounding air causing an ITT rise.
Although my engines are different, when starting a hot engine fuel is not introduced until the N2 is 20% AND ITT is 120 C or less.