While I was searching to identify this font, I hit gold and also discovered another restaurant on Rarotonga using the same one:
The word "restaurant" in particular - possibly also "Tumunu" but if so, it's been force-squeezed a bit horizontally. It does look very similar, it's just a bit too narrow.
I've used one of the Goudy variants - GoudyCat, I think - on a book project before, but they all look fairly different. This version here certainly wouldn't have suited my science fiction book!