Another research discovery
Forgot to mention another find. Remember, back in 1935, the understanding of the commerce power was that Congress could control, well, interstate commerce. How to apply that to a person who bought something locally and kept it at home and never took it interstate was a serious question. That's why the NFA 1934 was based on the taxing power. The registration provisions were supposedly just a tax return to ensure that tax had been paid on the firearm.
One memorandum, from the Assistant to the AG to the Atty General, suggested they might invoke the reference in the preamble to the Constitution to providing for the common defense. Thus they could register all guns with the excuse that they might someday be needed for the common defense, and thus the government must know where they were. He added it might even be possible to go farther and even seize some of them, using this power.