The temperature can be calculated without ever knowing the total power output.
For being precise the camera is nit sensitive to temperature but to radiated energy. For making a measurement you must know the ratio of the total emitted power from the body surface and that that would be emitted by a Black Body. So all your reasoning means that total emissivity is an important factor to know during a measure.
Total emissivity and not band emissivity because, as you demonstrated, that for many bodies the energy emitted in a certain band could be identical.