They will have your life history at that point and I think can enter your home to look around and search for it at any time with no warrant. I'm not 100% certain on that but am fairly sure. Not worth the hassle to me.
The part about entering your home is untrue. It's a popular urban legend, repeated often enough that people are "fairly sure" it's true because they have heard it enough times. If you have an FFL (a business license, not a C&R), they can enter the address listed as the place of business. That's the bit of truth that the "they can walk right in" legend is based upon. If they want to see your silencer, machine gun or short barreled rifle, they have to ask. They may indicate that declining to show it to them is 'a bad idea', but you don't have to talk to any law enforcement officers without your attorney. You can show them the gun, answere whatever questions without your lawyer, but that's your choise. Buying a registered weapon IN NO WAY waves your Fourth Ammendment rights. Buying a "class 3" item doesn't mean having a "machine gun license" it means you buy a registered weapon. There is a huge difference.
As for the part about them knowing your whole life's story, yes, they do. That's true regardless of wether or not you file an ATF form and pay the $200. Our lives are verry much a matter of record. Since the Social Security number has become a de-facto identification number, with everything from out credit records, tax records, and health records catalogued by that number, we have no privacy. If "They" want to know, "They" can find out. Should any one of us fall under scrutiny for whatever reason, the amount of information at the disposal of those who want to take the time to look it up is comprehensive.
If you want the silencer, buy it. It won't change your life in any way except that you will be able to shoot without hearing protection and your garden will soon be rabbit free.