When does an island with isthmus become a peninsula?
I think the naming difference is largely qualitative and even arbitrary in some cases. There are lots of 'islands' that are technically connected to a larger landmass by an isthmus of some sort, which would make them peninsulas technically, and not islands.

If you take the definition of a peninsula to be a body of land surrounded by water on three sides, then yes, the island in question could technically be called a peninsula. 

Local names of landforms are often linked to cultural or historic influences, just as much as they are to their technical description. There's no 'threshold' of isthmus-width (as far as I'm aware), where once crossed, a peninsula-like island becomes an island-like peninsula . If there was it would be entirely arbitrary.



As an interesting aside, the Isle of Harris in Scotland is not only not an island (there's an isthmus connecting part of it to a larger landmass) but even the isthmus part is not used to define the limit of the island, which occurs at some seemingly arbitrary point on the larger landmass! (see the map in the link to see what I mean)

https://en.wikipedia.org/wiki/Harris,_Scotland