Could the bridle be pinching her hear? I’d try a larger browband before anything else. 
I have one of the el-cheapo Nylon ones, kind of like http://www.horse.com/Western-Tack/Headstalls/Nylon-Halter-Bridle-with-Reins-WCM10.html
A lot of how much you like them depends on your horse, I think. Loved that bridle on my old QH. He was actually pretty sensitive, but out on the trail, he was happy and on a mission, and really trucked along on a loose rein, so I never noticed any problems with it. And it was fabulous when I went trail riding while I was pregnant, and had to stop to pee behind a bush every couple of miles.
My new mare is not so fond of it. It’s not awful, but I definitely feel like it’s a little floppy and sloppy, and she fusses a little about it when I have to package her up in spots.
Personally, I think it’s worth a shot to try.
What would be the difference between that and just putting clips onthe cheek piece of your current bridle?
I had a horse that was impossible to bridle too… the issue with him was his ears. He had such a phobia about it that even AFTER THEY WERE ACTUALLY HEALED, HIS FEAR OF GETTING HURT WAS OVERIDING everything. I cured him by putting snaps at the end of the cheek pieces, unsnapping the bit, setting the bridle on top of his head, then snapping the bit up again.
Took about 6 month for him to be completly totally fine. He now basically reaches for the bit , lowers his head for bridling…
(sorry about the caps - hit the button!)
I was thinking the same thing, but you need cheek pieces short enough to accomodate the snaps.
I also have a cheap nylon deal I bought from either Jeffers or Country supply. They work just fine, once you poke a few more holes so they can be adjusted properly. I use a small soldering iron to burn holes in nylon straps. We bought the halter/bridles so we wouldn’t have to risk having the horse get loose at the trail head when we pulled off the halter to put the bridle on. Now I have no fear of that, but at the time with young horse, it was a good idea. Endurance people use them so they can remove the bit at vettings to allow the horse to graze/eat/drink and still be able to use the lead. Also, they are better for tying, unless one likes to have a separate halter and bridle.
I also like to have clip-on reins. The OP might like this option as well so she doesn’t have reins dangling when she attaches the bit. You can get the clip-on pieces at some tack shops or hardware stores. You could use the same type of clip for reins and cheek pieces if you are looking for a short-term solution while you figure out what is bothering your horse so much.
