Due to the laws passed in the last 40 years, yes, the Director of the Treasury (head of the ATF) gets to determine what is sporting and what is not.
We objected then (and each time) but were out voted. Note the (in)famous Clinton AWB. It passed. We reacted, by voting as many of the supporters out of office at the next election as we could. And that did make an impression on the politicos, even though they were careful not to say so in public. In public, it was the Republican "contract with America" that caused the end of Democratic rule in Congress (for the first time in 40 years!), but in private, they admitted it was the AWB that cost them the seats.
We don't have much ground to stand on as a 2nd Amendment right when it comes to imports. They can, and do ban/tarrif/tax imports as they see fit. Legally that does not infringe our rights. If there were no guns made in the US at all, it would be a different matter, but its not.
We do not have a constitutional right to an imported weapon, be it a Saiga shotgun or an AK 47 (semi). Back in 68, the GCA banned the Walther PPK, because it was 1/8" too small in one dimension to meet the point system decreed as suitable for sporting purposes. Walther responded with the PPK/S, which was large enough to meet the arbitrary requirements.
Remember all those "assault weapons" with ugly thumbhole stocks? Again, done as a legal way around the Bush (the first) import ban. Anti's called it a loophole. The rest of us called it good legal business sense.
Heller basically says we have a right to our arms, in a personal sense, subject to "reasonable" restrictions. As long as you and I can obtain "suitable" arms from domestic makers, we cannot bring suit (or hope to win) against the govt for restricting imports. Sorry, but that's the way it is under current law.