The Constitutional arguments made in the source cited by muon are mush. It invokes
Hepburn v. Ellzey (1805) in a manner contradictory to the plain language of Marshall's opinion. Marshall explicitly writes: (emphasis added)
What
Hepburn v. Ellzey said was that Congress could grant access to the inhabitants of Columbia to the Federal courts in the same manner as it had to the inhabitants of the States and to foreign citizens, but that the law as it was in force then did not and that it was up to the Congress and not the courts to correct the anomaly as Columbia was not a state under the constitution.
Since Article 1 Section 2 Clause 1 clearly states that "The House of Representatives shall be composed of Members chosen every second Year by the People of the several States," Any court that upheld H.R. 5388 would have to overturn an over 200 year old precedent.
The next cited case
National Mutual Insurance Co. of the District of Columbia v. Tidewater Transfer Co. Involved whether when Congress finally chose in 1940 to eliminate that anomaly as Marshall himself practically begged Congress to so , whether it was constitutional. It did so, not by finding that D.C. could be treated as a State but on the use of other powers, that were not dependent upon D.C. being treated as a State under the Constutution.