The Constitution says Article I Section 7 regarding the authority of Congress "To exercise exclusive Legislation in all Cases whatsoever".
Since Congress didn't enact the legislation per normal procedures outlined in the Constitution, the law has no Constitutional authority.
I predict the Supreme Court will get to that part and make its decision per the words in the Constitution ~ and that's the end of the DC council's "gun law" and the start of DC government's Executive Branch "gun rule".
The Supreme Court has held that “exclusive” simply means the ceding states would retain no authority over the District; it does not mean that the power is nondelegable.
Check out District of Columbia v. Thompson Co.