The bed sharing isn't the main issue.
I wouldn't be comfortable with it especially in such a new relationship. It would concern me that one place the DC might mention any concerns they had would be school - that's less likely as the OW TA is already talking to the DC at school - and the Dad works at the school. I'd be worried that limited her options if there were future problems.
Big issues - a person in position of trust encouraging a child to lie.
Big thing - I personally come down hard on GP when they've tried that. If DC is unlucky to encounter a predator if they've already been taught to keep secrets they are more at risk - and two people working with DC should really know that is unacceptable more so than average person.
Bad mouthing the mother to the child on school premisses.
Bad enough but OP states another TA over heard this and spoke to HT already. It's got back to the mother as well. So it's a source of gossip.
It's possibly human response but it's very unprofessional and unfair on the child. It very stupid on TA part anyway - any issues with other DC and at my DC school those parents would be pointing out the poor judgement and lack professionalism of the TA and wouldn't have a bar of them.
Does the op trust the ex to protect their dd?
I image at such a raw stage trust is minimal. I expect OP trusted her DH not to cheat on her, I expect she trusted he'd put their DC first which doesn't appear to be happening.
Frankly if nothing is done I expect the OW TA would continue to bad mouth the mother.
I won't tolerate teachers and TA bad mouthing me to my children - I'm surprised the OP is expected too.
I don't think social workers would be interested but OP has done the obvious thing of talking to the father - with apparently no impact so next step is asking school for some help managing the situation better.
I suspect a quiet word to the TA reminding them they are under scrutiny by parents and need to behave professionally would be first step though I don't know but the HT would know. It could just be to keep a closer eye on the situation by other staff.
If ex and OW TA get pissy - I point out of they had behaved more professionally at school there would be nothing to comment and that their DD needs support in this difficult time.