I wish her mum and I could be OK with each other but there's such an almost tangible feeling of hatred that comes from her when she's near me. I know my SD picks up on that but is that then for me to just disappear into the background.
It's about what your SD needs - what she and you want isn't necessarily the right thing for her long term.
The best thing I ever did with DHs DCs is let them know that I know how their mum feels about me, and that I wasn't angry about it. I refused to allow it to be the "elephant in the room" that everyone knew but no one talked about.
A lot of their emotional conflict came from embarrassment that their mum didn't like me, and then feeling guilty because they felt embarrassed by their mums feelings about me. I took away the embarrassment by openly talking about it as something that just "was" - I made no judgements about it, just accepted that was how life is. For instance, we turned the repeated implication from their mum that I was a wicked stepmum into a lighthearted term of endearment!
I also made sure that they knew that I would understand if they withdrew from me because of their loyalty to their mum.
Sadly, perhaps as a result, they have both chosen to abide by their mums wishes and have repeatedly cut contact with me for months/years at a time. however, they know that I understand why, and both of them, at different times, have reinitiated contact in the knowledge that I'll accept them back despite their rejection of me. They are, as a result, emotionally healthier than they would have been if they'd been forced to continue to live with that emotional conflict every other weekend.
It's harder now that DHs DD is an adult - I'm not sure for how long I can 'excuse' her behaviour as loyalty towards her mum, but we'll have to see what happens if when she reinitiates contact again.