Curonians were named as a people at least as early as the 12th century.
Wikipedia cites the Norna-Gests þáttr saga:
It's also referring to events in 750, so it's quite possible that the name goes at least that far back.
I'm not quite sure where you got the notion that the notion of the Curonians being Finnic is "the most popular hypothesis at this point in time." I'm not very familiar with the topic, but the little bit of research I did seems to show that the view of them as Baltic is quite dominant, with a very small minority believing they were Finnic.
(I'm talking about scholars here - I have no idea what the man on the street believes, nor does it matter.

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