The law is basically thus:
Private citizens can only own pre-1986 registered fully automatic weapons. These are called transferable weapons; for ownership to change hands a background check through the BATFE and $200 tax stamp are required PRIOR to sale.
Since the supply was cut off by law in '86, prices have shot through the roof. Complicating matters are what, exactly, the BATFE considers an NFA weapon. In some cases, just the sear for a FA gun is the NFA "gun". In some cases, a receiver (or part of a receiver, such as an AR/M-16 lower).
Further complicating what you want to do is the fact that 3-round burst fire is technically fully automatic fire. More than 1 bullet per pull of trigger = FA.
Since your AR was built as a non-NFA gun, adding the fire control parts to make it one can be done two ways. One way will land you in prison - that is to just put the burst trigger parts, sear, safety, BC, etc. that are needed to make the gun fire full-auto. The legal way would be to buy either a transferable drop-in sear or M16 lower on the NFA market (background check, stamp, etc) and assemble it into what you and I would think of as a "gun" using the necessary parts. As supply has dwindled, you may end up spending $10,000 on an AR sear - a very tiny, simple part. But to the BATFE, that's the FA gun.
IMHO, real FA guns are more of a collectors' toy than anything. They will appreciate in value, but they're too expensive and restricted to shoot casually. Using one for self-defense would be a legal nightmare, even if justified. Hunting with FA is not allowed in most places. Most shooting sports don't allow FA (there are actually a bunch of exceptions to that fact, but competitively shooting FA is a well-heeled sport... akin to privateer NASCAR racing).
If you want 'sorta' FA, but legal, look into a slidefire stock (or how to bumpfire). They're fun and legal for now; might not get the cyclic rate on a real FA, but close enough.
IF you're really intent on owning NFA firearms, there are ways to reduce costs that mostly involve going into business making or selling them, with all the attendant costs of starting a new business. The BATFE looks at these folks pretty closely, so if you're thinking of running a 'sham' business (hobby business to them) they will pull your license, and your NFA assets have to be rather quickly sold to another dealer or destroyed.