I was afraid that this was going to happen. Frankly, I'm surprised that this hasn't happened sooner, and to more of us.
If you don't meet two of those three criteria, you're screwed. You can be convinced that you are right, but they are the ones who need to be convinced, and it seems that they aren't.
You need to hire a lawyer unless you plan to pay back all of the money. At the very least, get a CPA who is an Enrolled Agent. That is someone who is permitted by the IRS to represent you. He/she can submit what is known as an Offer in Compromise. This is basically you telling the IRS that you can't afford to pay what they say you owe, and offering to pay them what you can afford.
Unfortunately, you need the lawyer if you plan to get the ruling overturned. It will cost you thousands of dollars in atorney's fees, and take years. If you lose, they will make you pay the full amount plus interest that will continue to accrue while you are fighting in court. The court, by the way, will start with an administrative law judge who basically works for the IRS. You have almost no chance of winning there, but your lawyer will still charge you while wasting your time there. Your best bet, I'm afraid, is to make the offer in compromise and hope they accept.
The other thing you can do is work it off by turning in your fellow techs to the IRS. They'll give you a cut of whatever they collect from them. If you choose this route, I'd recommend finding a new line of work too, since you won't exactly be popular among all the coworkers you screw over.
The next thing you must do, as should all of us, is ensure that you meet those criteria for having a tax home in this and all future years.
It is all too common in our business to do our taxes by the alternative "urban legend method". Some of us do things like write off the difference between GSA rate per diem and the per diem we got as if it were a deduction. Some of us work a long term job for 11 months at one place, take a month off, and then go back all the while collecting per diem. There are other dumb things too, but these are the two biggest offences.
Someday, it will become common for travelling nuke workers to actually do taxes according to the federal regulations that are in effect, but it that doesn't get us back as much money as the way some HP told you you could do it.