Is she safe to just ride out somewhere? Like a nice trail ride? Maybe take her out in the real world, ride some hills, negotiate some terrain?
I've got a barn-sour, hyper-active mare but I never actually lunge her. I tried last winter but it didn't seem to do much good. She just got more mentally stimulated. Now I just ride her out and go to-and-away from home until she finally settles. In the summer when I ride her consistently she settles and walks politely whenever I want.
In the winter when she sometimes sits for 2 weeks due to the weather, we have to start all over again, but hey, it works for us.
Sometimes also people and horses are just not a match. I sympathize with your plight. I have a long yearling that is driving me bonkers. I don't know if he is going to work out for me or not. So I understand. But if you love her and hyperness is her main fault, I think you can ride that out of her.
You might try leaving off the grain. I know you said she needs to gain weight/muscle? I would try a higher protein feed (maybe some alfalfa) and/or a fat supplement. But carbs she probably doesn't need. Although honestly my mare is still a basket case even without grain, so I still give her a little bit anyway.
How much grain does she actually get? I feed my mare 1-2 pounds a day. If you feed quality hay with quality protein then she may not need grain to maintain her weight. I don't think the grain is causing the problem, but it could be making it worse, you never know.
As for the head-up-thing. Have you tried a different bit? Tried less bit contact? It sounds like she really dislikes the feel of the bit, either due to the bit itself or tooth problems. I don't know about your horse personally of course, but my horses tend to throw their heads up in a normal snaffle. You could try one with a 3-piece center or a bit with a solid mouthpiece or a mild hackamore to see if you get a different reaction.
I ride western and I mostly ride in leverage bits with little contact, so we are comparing apples and oranges here, but don't be afraid to try different bits to see if you find one that aggravates her less. I change bits all the time depending on my mood and how I plan to ride that day. It wouldn't hurt to dry some different bits. The one you are using just might not work for the shape of her mouth. Or maybe she is just so sensitive that she can't handle all the contact. Who knows. Maybe try something bitless to see if it's her teeth/mouth pain.
Best of luck, I feel for you and hope you make some good progress soon.