OK, whoa back a minute maybe, she has only been started 4 or 5 months and she is hard headed? I have a red flag waving, something isn't right with this set up, especially of you have tried all those bits including a twist on a greenie.
First thing as Toto says, check the bit is actually fitted right, I have one mare who plays with her bit terribly if the headstall is even one hole to loose.
Once you have a bit and a headstall that you are sure fits her, and is fitted properly, then back to basics with her training to soften her up again.
Personally I would stick with the French Link, just because I like them, I think they are a great starter bit. Don't worry yet about what she may wear in the future when you start barrels with her, you need in this case for sure to be able to walk before you can run