I would check out the full-auto forum or legal forum.
It sounds like you are confusing two separate things: 1) The single transfer tax stamp and 2) the Class III dealer's license.
Part 1:
Anyone with a clean criminal history who lives in a Class III legal state and who has the permission of their local sheriff can buy a Class III (full auto in this case) firearm made before 1986. You go to a Class III dealer and buy the firearm (made before 1986) that you want. This will probably cost at least $4,000 to $10,000 for a M16 type weapon. (The high price is from the dwindling supply of pre 86 full autos.)
Then, you don't get to take it home after you buy it. You pay all that money, plus $200 (I think) for a "Tax Stamp". I’ve heard that some dealers charge something like $50 for all of the paperwork too. Then you wait many months for the paperwork and background check to clear the BATF. Then, assuming you pass, you get a slip of paper with your Tax Stamp on it. (I think the tax stamp looks like a big postage stamp.) At this time you can go to your Class III dealer and take home your new toy, assuming you got the permission of your local sheriff.
As far as I know there is no law that allows the BATF or anyone else from coming into your house without a warrant. You must at all times keep the Tax Stamped paperwork with the firearm. If caught without the papers, you are in big trouble. When you take your Class III out shooting, you must bring your papers.
If you sell the firearm, the person who buys it from you must also go through the tax stamp crap and pay $200 and wait forever and a half on the BATF. (I think a family member can inherit the weapon without the charge for the tax stamp.)
The law that created the tax stamp was passed in 1934 (I think) because of the hysteria stirred up by the Thomson Sub-Machinegun (A. K. A. “The Tommy Gun”) and it’s use by a few high-profile gangsters. ($200 for a tax stamp was a lot of money in the 1930’s. On the History Channel, I saw ads before the tax stamp law that were marketing the Tommy Gun to ranchers and such as a tool of self defense.) Up until 1986 you could buy newly manufactured full auto weapons (with a tax stamp), when they passed a law banning post 1986 full auto weapons from private ownership. I’m not sure, but I don’t think the ban applies to silencers/suppressors (which require a $200 tax stamp of their own).