I have an all-new air conditioning setup that was installed in April of this year. (I mean ALL of it -- heat pump, air handler, thermostat, and ducts. The house previously had air conditioning, but the condition of the existing equipment forced us to replace it.) I am having an issue with the heat pump, which is an American Standard 2-stage 3-ton split 18 SEER 13 SEER, according to the documentation I have. For two months after installation, it ran perfectly, and it was amazingly quiet. Even running in stage 2, it didn't make very much noise.

Then the noise from the heat pump started getting louder. It makes an exceptionally loud twanging, buzzing sound whenever it turns on, and when it runs, it makes a loud buzz, on top of the normal sound of the compressor. The buzz is constant, and mostly constantly loud, but it can get quieter or even louder. Also, the "whoosh" sound that the compressor makes when it shuts down seems to have gotten louder, and is preceded by a grinding, loud buzzing noise, and a "thunk." Often the AC will shut off after making that noise and immediately turn back on -- and it will make that loud, twanging buzz when it does it.

We have had contractors out to look at it twice, and try to fix it, and they have told us that the compressor is running fine, with optimal pressure, except for one thing: an electrical part (I think they said the low-voltage regulator) is physically vibrating and making the buzzing noise, which gets louder or quieter depending on the amount of current going through it. They said they did not know why this was happening. They replaced the low-voltage regulator, and the noise persists. The AC contractor does not know why.

This noise has rendered one bedroom in my home unusable. The constant loud buzzing carries through the whole house, and it makes it very hard to concentrate or relax. Please, can someone help me figure this out?

The AC contractor does not know why.
Then it is his business for find out. I never heard of a voltage regulator on an ac or heat pump. You used the term heat pump repeatedly. Do you really mean heat pump or do you mean condenser unit. Is this an ac or is it a heat pump (ac and heating) system.
Sounds like a faulty contactor. But that is relatively simple to determine. If the unit is a heat pump and the noise occurs only in heat mode it could be a faulty reversing valve.
Regardless it is the installers problem to diagnose and make right. If you can't get satisfaction from installer call manufacturer.

It is a heat pump, capable of heating and cooling. So far we have only ever used it in cooling mode. (We will not need to heat the house for several months.)

I agree that the installer should figure it out, since the thing is under warranty and he did the work -- but when he says he doesn't know why that particular electrical part is making the noise and he doesn't know what to do, then I'm not sure if he will figure it out.

The contact the manufacturer through their customer service line.