Out of all possible nicknames you can get out of Elizabeth, this is probably the worst one. It's worse than Buffy! In American pronunciation, it sounds just like Teddy. In British pronunciation, it sounds a bit like 'titty'. And it looks like it in both British and American. This is an ugly and silly name. It's quite infantile as well.
Elisabeth Johnson was the wife of famous Enlightenment writer Samuel Johnson, and he often called her "Tetty" or "Tetsey," as those were common (if rather girlish) diminutives at the time. James Boswell, Johnson's good friend, affectionately razzed him about it a bit when he did his biography, "The Life of Johnson."