Is FDA registration required to be tested by a designated accredited laboratory?
Nonexistent...
FDA is an enforcement agency, not a service agency, so there are no public service agencies and laboratories, nor are there "designated laboratories."
FDA, as a federal law enforcement agency in the United States, will only recognize GMP quality reports issued by service testing laboratories and issue certificates of conformity. Certification bodies are not "designated" or "recommended". It's like "the referee can't be a player at the same time" : he can't whistle and kick the ball, and he can't designate a goal -- otherwise it's a black whistle.