In a word, No. The F12F designation was assigned to a derivative of the F11F Tiger that was to be powered by the new General Electric J79 engine. It's not clear if a contract was ever issued but two Bureau Numbers were assigned. The program was canceled in January 1956. The effort to evaluate the F11F with the J79 continued, however. The first of two F11F-1Fs flew in May 1956.
Pictured is the Grumman Design 118 proposed to the Navy in late 1955. It and a single J75-powered alternative were rejected in mid-1956 in favor of the Vought XF8U-3. It is unlikely that the Navy ever considered it seriously enough to assign it a designation since the McDonnell F4H was already on contract. Somebody subsequently and erroneously concluded, because of the concurrency of the projects, that it was the D 118 that received the F12F designation. Someone else speculated that it would have received the popular name "Lion". Officially it never did and I've seen no evidence that Grumman referred to it internally as that.