sea shanty is also spelled chantey (similar to chant and chantez, chanson, etc). I pronounce both spellings the same.
The derivation (from French or from Latin) could be indirect; we already had the word "chant" in English, but that wouldn't explain the pronunciation of the sh/ch sound as in chanson but not as in chant. Or maybe English speakers used to pronounce chant more like the French?

For what it's worth, an exhibit at Ellis Island states that the word was derived from "chanson". Or did when I visited ....

I have only come across one spelling for the house type shanty - until I saw the lyrics in this thread where it seems a pun has been made deliberately or otherwise to form "Chanteytown". Q writes, "Shanty, for a shack (small log cabin at first) probably came from Canadian French chantier, a logging camp hut. It appeared in the 1820s in print. It was noted, however, that those living in shanties were often Highland Scots, Americans and Irishmen (McTaggart, 1829, 'Three Years in Canada')." As the Highland Scots and Irishmen would have been Gaelic speakers, I like the theory of "sean tigh", meaning "old house", as the derivation in this case.