If you google "algy met a bear" you will get thousands of hits on the traditional "Algy met a bear, the bear met Algy. The bear was bulgy, the bulge was Algy". If you look up pre-1940 _bulgy bear_ in Google books you get hits, such as this one in Everybody's Magazine with some version of a bear meeting Benjy, with predictable results.
Not that our Narnian bears would do such a thing: I'm sure their bulge came from eating too much honey.
From which I conclude: when _Prince Caspian_ was written, "Bulgy bear" was a known thing, appropriated & repurposed by CSL.