Whatever issues she has at a run can't be improved until the slower gaits are improved even more. Even if you think she can lope the pattern perfect - go back and perfect it even more. It makes a huge difference. Don't run her for a while and just focus on it. It sounds like she has issues breaking at the poll and resisting the bit/hackamore pressure. Here's the bits I would definitely have to say put her in a full cheek snaffle. I use a slow twist full cheek on mine when we work on these things and it's amazing. Do tons of circle work just getting her to shape to your leg and bit pressure. Use your outside leg to keep her in tighter circles. Hold the reins with two hands, you will want to spread them apart onto each side of the horse. Lift up your inside rein and lightly pull for her give into it while pushing her in a circle with your outside leg. As soon as she gives to the bit release. Eventually she will start getting a beautiful headset. Yes it is just as much important to get a horse that can achieve a good headset in speed events as it is in say a pleasure class. In order to make a good arc for your axis points she must must must be supple from these exercises. Once she can do these circles like that add a barrel in the middle and don't try and have her do patterns. When you believe she is consistently keeping that arc go through the barrel pattern. Make extra turns around barrels to check her. This will prevent her from rushing out of it and keep her listening to you. Keep things slow. If it doesn't work at one speed it is definitely not going to work at a higher and less easy to control speed. Oh and if you get her using her body for those circles and such she won't be running through the bit. If she even attempts to keep circling her and she definitely can't run through it then. My paint/arab has a mind that is only in the fast lane during a barrel run. Originally I claimed she ran right through snaffles and I only rode her in her sweet six gag bit. We then went back and worked on this. Now you can ride her anything even a cowboy halter.
I am a hackamore hater. I've seen more things wrong with them than good. When my pleasure gelding had been abused by someone at a previous stable he did not wants anyone touching his mouth. We got him an Indian Bosal off of eBay
and I worked him in that for awhile. With lots of consistent work he is now riding in just a snaffle again. I don't show him anymore and he is super light to ride so he doesn't need anything more. Honestly though, if your mare doesn't have any previous oral abuse and you've had her teeth floated(I suggest this before you do anything as it can be the number one issue for head tossing) then find a bit that works for her and there are so many different available ones there has to be many she likes. Even if you don't run in them all the time it's a good quality for your mare to posses.
ETA: Don't add a tie down unless you absolutely have to(i.e. My mare used to spook and automatically her head came flying into your face bout to knock some teeth out). Many I've rode and trained had been claimed to need it but they did even better without their face being tied to their chest, imagine that.
Oh and tom thumbs...there isn't a bit I've used and disliked more! It's nothing more than a poorly shaped broken curb and dampers training. But that's my opinion of course.