Indeed the temperature of an object would be hotter being by limited to IR output in a spectral band portion of the IR band than it would be if it were a blackbody at the same temperature.
I said that a bit sloppily.
Let me rephrase:
Indeed the temperature of an object would be hotter being limited to IR output in a limited spectral band portion of the IR band than it would be if it were a blackbody at the same level integrated radiant power.
In fact I tested and reported on this effect a long time ago. I tested a frosted silica tube, then painted it black, using the same input power for both. The temperature dropped, and the emissivity went up when black.