Oh sweetie. I think you have to get her checked out (her ears I'm thinking) just so you know feel reassured in yourself that she's not in pain. Although if she was in discomfort, it doesn't make sense that she would settle easily with a BF? That's what makes me wonder if this is an independent sleep issue.
If you decide that this is about independent sleep I think my plan would be to be 100% strict about BF to sleep. I would BF before naps and at bedtime BUT: at nap I would BF downstairs 15 mins before starting wind-down. That way if she kicks off asking to be fed you know she's not hungry (like, if she hasn't eaten well at lunch) and she's had the cuddle time too. Likewise at bedtime I'd introduce something between end of BF and bedtime. Like, brush her teeth, or do a story, or BF her in just her babygrow and put her in her sleep sack after BF. So she's properly awake.
Can you talk to her about it at the start of wind-down .Real simple language but something like "you're going to go to sleep now. Mummy will stay with you in your room but it's time to sleep." So she knows what's expected. I wouldn't shush or touch her, I'd just sit next to the cot and say "You have to lie down E, it's time to sleep, mummy is here." Matter of fact. Does she have a lovey? I know it's late in the day but could you persuade her to take a dummy? If BF is comforting to her, maybe a dummy and the sucking action would fulfil the same need? Audrey gets a right head of steam on very quickly if she wakes up fully and I feel like it helps me to be able to tell her what to do. "Aud, grab your lovey. Find your dummy. Thaaat's right now lie down please. No, no fuss, no crying, it's sleep time now." I find that if I shush, she can just tune me out and screams so loud she can't hear me but if I talk to her (quite a loud, firm voice, not stern, but not whispering, yk) she sort of has to pause even just a tiny bit to listen to me. The good thing about talking to her rather than shushing or touching her means I can do it from right the other side of the bedroom if needs be.
Also with the signing, can you teach her the sign for 'hurt'? WHERE and HURT are so useful, if you can ask her, then perhaps she could show you where/if it hurts?
I'm so sorry you're all having such a rough time