Hi & welcome to HF,
I don't know what btv means... bit to violence??
It sounds very much like your horse needs some good training, has just been forced & hung onto with the bit instead of being *taught* to *yield* to it. Sounds like she's new to you? I would actually forget events & such for now, and work on getting to know her & training her to respond to you well first. But first & foremost, I'd want to be thorough about ruling out/treating any pain/discomfort that may be a big part of the probs.
always been strong, I can hardly manage her on the flat in a snaffle indoors, let alone outdoors. Her previous owner never worked her outside unless they were showing, creating a huge stigma
That she's always been 'strong' means that she's resisting the bit, not responding to it. If she's a lot better indoors, that's where I'd start, and I'd get her going well on the ground first before riding. I'd also get her going well at a walk, before attempting trotting, etc.
She has a fair bit of nerve damage to her tongue, and when I pull too much she gets very heavy in the bridle.
Recently she’s been extremely hot, so I shorten my reins like any unknowing rider would.
If she has nerve damage to her tongue(what from??), it's hurting her, I would NOT be riding her in a bit. I would ABSOLUTELY not be putting any significant pressure on the bit. There's one pain factor you may be able to address easily.
Not sure I get why you shortened your reins or pull 'too much' if it sounds like you know better. Are you a learner? If so, as it sounds like she's a 'difficult' horse because of past treatment, I'd consider finding a good rider/trainer to work with her, rather than you doing so.
She may be 'extremely hot' lately due to too much feed, not enough exercise, etc. But I suspect she's 'hot' with frustration at not being able to work out how to get out of her pain.
I’ve tried long rein trotting in attempt to get her nose poking out but that was no good and made the problem worse.
Yeah, sounds like she is not able to relax under saddle. You need to address her *attitude* about being ridden not just her nose
Obviously if she's hurting/worried about hurting, you need to deal with that first & ensure it never happens, for her to learn she doesn't need to stress about it. Unfortunately it depends how 'ingrained' that attitude is as to how hard/easy it will be to 'fix' even when the pain is gone. If she associates the bit with anxiety because of tongue pain, then going bitless will help avoid the previous associations too.
As said above, I'd start at a walk, not trot - maybe even at a standstill if she's anxious there too - and teach her to *relax* at each pace, before asking anything else at all. Then you can work gradually forward to whatever goals, but be able to fall back to a Good Place if you find you've gone too fast for her & she's getting antsy.