You are looking at something that weighs less than mount Everest and thinking that's an unrealistic amount of mass to control.

In actual fact, moving mount Everest To The Moon is trivial in comparison with the task of creating a black hole. It's a feat only accessible to Kardashev type 2, at the very least 1.5, civilisations. We are talking about civilisations that can move small planets at will.

The most credible approach is to build giant gamma-ray lasers, and focus so much energy into a spot to small, that the weight of light itself will collapse the spot into a tiny black hole made of pure energy, Kugelblitz.

Once you would have a black hole formed that way, you could continue force-feeding it with lasers or particle beams to weigh whatever amount you desire. It would be a very energy-inefficient process.

At some point the black hole reaches a mass where it's Eddington limit allows it to feed by 'normal' means faster than it evaporates by hawking radiation. At that point it could server to convert matter to energy.