My uncle works for Bath Iron Works in Maine and has been telling this factoid for years. It seems like an incredible achievement. Are there any resources to back up his claim?
No.
Battleships took around 2 years to build + several years of R&D in the United States.
They might have been able to assemble a liberty ship from per-fabricated parts in that time. The record for such a construction was the Liberty Ship SS Robert E. Peary built in 4 days, 15 hours, 29 minutes. You'd have to check and see if Bath built liberties, but most likely they were involved somehow.
Here some more information on the SS Robert E. Peary.
They didn't launch a battleship every two weeks - the Bath Iron Works never produced a modern battleship. They did produce destroyers, which are much simpler ships to build. During the ~100 week period between February 1942 and November 1943, they launched and delivered 32 Fletcher class destroyers, roughly one every three weeks. Looking at their list of delivered ships, there are several pairs delivered two weeks apart - the USS Norman Scott, delivered on the 5th November 1943, and the USS Mertz, delivered on the 19th are one such pair. This doesn't mean that they took two weeks to produce, just that the average time between completions was two weeks. The yard had several slipways, and ships were worked on in parallel. The list of ships can be found on the Bath Iron Works site.