Very Unlikely (without an amendment)
SCOTUS has before rejected the idea that Congress can fiddle with the internals of SCOTUS itself beyond the specifically prescribed constitutional powers of "advice and consent" on appointing justices, impeachment, and controlling their appellate jurisdiction. Laws on ethical behavior and recusal for federal judges has, for example, been rejected as applying to them. You may recall that before his confirmation to SCOTUS, there was an ethics review being done on Kavanaugh as a result of the allegations of Dr. Ford. The board has legal authority to conduct such reviews over all federal judges, but after Kavanaugh's confirmation his ethics review was quickly dismissed for lack of jurisdiction: the board simply had no authority over SCOTUS itself. This is done on the basis of the separation of powers—the justices/SCOTUS form a separate and co-equal branch of government to Congress and the President—, and that the constitution is intended to be one of enumerated powers, and there is no power enumerated for determining the structure of the Supreme Court.
A peculiar exception, perhaps, is the membership of the court itself. Nothing in the constitution states how many justices there are to be on the court, and the position of "Chief Justice" is only brought up when describing the impeachment trial of a President in the Senate. There is, in particular, no specifically ascribed power for Congress to determine the membership of the court. It has, to date, been traditionally ascribed as an implied power of "advice and consent" to appointments to the court. Limiting the number of members can be construed as a pre-requisite to consent: currently, if there's already 9 Justices, then consent is denied. Congress could change the threshold at which it automatically denies consent by simply passing a new law. There are some who believe that SCOTUS is the true holder of the power to decide the number of its members, but at this point it would be a very radical departure from traditions, including multiple changes in the court's size by Congress over the nation's history that were simply accepted, if the justices actually tried to assert such a power. And as appointments to the court remains specifically a power of the President and the Senate, it's questionable what practical effect they could actually give to such an assertion. An easier-to-exercise claim to power would be to assert that SCOTUS decides who their Chief Justice is, though this would again fly in the face of a tradition that has lasted for the nation's entire history.
It is SCOTUS itself which has decided what role the Chief Justice actually plays. For the most part, the Chief Justice is equal to each Associate Justice, and the Justices make various decisions democratically, though by tradition the Chief Justice decides who writes the majority opinion (and it is frequently written by the Chief Justice if they join that opinion).
Another—I'd argue less significant—potential exception is how Justices are assigned over federal circuits. Congress has, via Article III of the Constitution, the specific power to create (and remove) and regulate all federal courts inferior to the Supreme Court, and the assignment of Justices to oversee particular circuits is done in accordance with Title 28, United States Code, Section 42. But it is SCOTUS itself which decides, through whatever process it so pleases, which of its justices are assigned to which circuits. That there can be a structure of justices assigned to oversee circuit courts is derived from Congress' power over the inferior courts, but the specific allocation of those Justices to particular circuits is, by separation of powers, left to SCOTUS itself. The law in question does specifically state the assignment is left to SCOTUS to decide, I will note. This was done as a recognition of the separation of powers, and a corresponding caution to the possibility that the law might have been stricken down entirely if going any further in specifying the assignment was deemed unconstitutional.
