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. The program was canceled in January 1956. (It's not clear if the contract was ever issued but two Bureau Numbers were assigned to the F12F.) 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 subsequent J75-powered proposal 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.