The Constitution says the Congress may prescribe the laws. That means the states must allow the Congress to set the process if they're going to carry out an act.
Congress may prescribe the law to prove or authenticate a legislative Act or judicial Act of a State. This means that Congress sets the process by which state authorities prove a submission to be an authentic copy of a State Act. Congress did set the process. At the time of the WBTS, it had been set by the Act of May 26, 1790. Currently it is set by 28 USC 1738.
This is the official opinion of the Unites States Legislative branch by way of its Act of May 26, 1790, and of the United States Executive branch by way of signing the legislation, and of the United States Judicial branch by way of the U.S. Supreme Court decision in MILWAUKEE COUNTY v. M.E. WHITE CO., 296 U.S. 268 (1935).