There is a book about the POR, from the blurb for which ChatGPT appears to have sourced its answer. However, I'd dispute the origin of the name given there as being RAF, since Churchill's famous speech to which it presumably refers is known as 'never surrender', rather than 'never give in'. Green's Dictionary of Slang has 'Bash on Regardless' though:
To bash on regardless is a phrase first recorded in 1948 and is an extension of bash on, meaning 'to persevere; to pursue a course of action regardless of difficulties' (OED). The word bash comes either from the Swedish basa, to baste, whip, flog, lash, or Danish. baske, to beat, strike, cudgel and its basic standard English meaning is to beat or hit. It has, however, been suggested that it might be onomatopoeic. The term seems, albeit unrecorded at the time, to have been a British military coinage, used in World War II. Thus this example in The Royal Engineers Journal of 1949:, concerning the post-D-Day invasion of northern France
Often frozen and wet through by night, they proceeded to march and fight at top speed for days on end. This was the 6th Airborne, with wings on its feet as well as its shoulders, and a spirit which said, 'Bash on regardless.'
During the 1971 India-Pakistan War, the Indian Army named one of their tanks, which was celebrated for its successes against the enemy, Bash On Regardless.
Merriam-Webster gives several vernacular uses for bash, but all marked as 'British, informal'. So maybe the word wasn't in common US usage even in the 1940s and would just have served to confuse?
There's a nice typo in the blurb for the book, BTW!