I believe black holes are not currently emitting any Hawking radiation, because the CMB is far too hot for now. They will start emitting Hawking radiation in the far future, after the Universe has cooled down significantly.
It's like standing next to a campfire. Your body is still emitting thermal radiation in the IR, it's just that you're receiving more thermal radiation from the fire than you're putting out.
This obviously is not a problem if you create a new very small black hole from scratch as it only applies to pre-existing black holes that are massive enough so that their hawking radiation temperature is lower than the CMB