There's still a shot to get the House to reinsert the full repeal as they now have the option to pass it as presented or add in their requirements and send it back. That's the reconciliation process.
So make the phones ring and keep up the pressure.
Okay, a brief synopsis of the reconcilition process and Byrd rule as I understand it.
First, two things.
All tax bills must originate in the House period, the Senate, cannot per the Constitution, initiate any form of a tax.
The filibuster remains intact in the Senate and any legislation, such as an independent repeal of the NFA or any portion of it seperate from the tax implications, can be blocked by filibuster. It takes a two thirds vote to end a filibuster and proceed with any bill in the Senate. So in effect the Senate is functionally unable to pass any legislation on their own unless two thirds are in agreement.
Now for reconciliation.
This is a Senate slight of hand that allows the Senate to pass their own tax law piggybacked onto a House tax bill. IMO, this is unconstitutional as tax law must originate in the House not added in later by the Senate. I would argue that the Senate can only vote yea or nay on a House tax bill by a strict reading of the Constitution. But then you have the filibuster so we have the reconciliation process that is immune from being filibustered and only requires a straight up 50+1 vote. Under reconciliation the Senate can remove portions or add their own portions to the House bill then send it back to the House for approval or disapproval. If approved it goes on to the president. If disapproved, it's dead and back to square one. The House can make their own changes, taking out Senate changes or restoring items the Senate removed and sending it back to the Senate. This process can go back and forth until one bill they both agree on is passed.
Now for the Byrd rule.
The Byrd rule, applying to the Senate only, says that the tax reconciliation process can only be used for tax related purposes only. The person responsible for keeping bills on track and on topic is the Parlimentarian, an appointed position determined by the Senate majority leader. The Parliamentarian is a glorified secretary, nothing more or nothing less that serves as an umpire calling balls and strikes on what can or cannot be in a bill or how the voting process is conducted. In this case, it could be argued that the Parliamentarian overstepped her authority in removing the registration portions, though the case could also be made that she split the baby.