Gibb, the OED gives no support for that pronunciation of "chaunt."

It lists it simply as an 18th and 19th century spelling variant of "chant," with the "ch" in "church" and the "au" as in "palm."

This suggests to me that if "shanty" had come directly from "cha(u)nt," it would almost certainly have had the "hard ch" from the very beginning. But if it had, I doubt anyone would have suggested French as an origin.