In theory if you used the power and radiated it to space (via laser, RF, etc) you’d be ok. You could even use also use that energy to put into rotational energy to increase the earths rotational speed

Lasers can't radiate heat, as a laser beam has no entropy. Ditto for coherent radio waves.

Fire a 1GW laser at the moon and you will transfer the best part of a billion joules of energy from the earth to the moon every second.

Sure. And you will transfer no entropy, so this cannot be used to cool the Earth.

SF authors (like David Brin) have screwed up on this very topic.

Lasers can be used for cooling, by shining a laser on a target carefully set up so that anti-Stokes scattering carries away entropy. The light scattered is quite incoherent, though.

You have to be very careful of Maxwell's Demon when it's a question of entropy: how efficient is that laser? How does the efficiency change when it is in a heat bath of X kelvin?

There may be some neat tricks, this is an area I know I'm not good at, but you have to be very careful.