OP I know you weren't asking about the bullying - but the next time this child bullies your dc and they're hurt (mentally or phyiscally) go into school and say that they are repeatedly failing in their duty to safeguard your dd. Tell the mum of the other bullied child to use this phrase too. Follow up the meeting at school with an email, ensuring that you use the phrase plenty of times - and that include what you want them to do to safeguard your child.
I know that telling school that they are not looking out for her and she is being bullied should elicit the same response as telling them that they are failing in their duty to safeguard her from xxx child - but one of them is just a set of words that they can ignore or brush under the carpet or act on in their own time. The other - duty to safeguard - carries a legal requirement that they need to follow up on, act upon and will get noted in stats and ofsted and the like.
If the other child's mum also complains of their repeated failure in their duty of safeguarding her child, when it's the same child causing the problems, it's going to be taken even more seriously.
you can ask what they intend to do to safeguard your child and hopeully they will come up with some good suggestions. However, you need to think it through in advance and have some of your own too (maybe in collaboration with the other mum?) rather than let the school suggest a few feeble measures.
If it's bad then it's worth suggesting a few strong measures - one - to show how seriously badly it's affecting your child - and two - because from a negotiating point of view it's always sensible to ask for more than you want so you can be negotiated down to what you really want! For example - you could ask that the other child is moved into a different class (don't accept them moving your dc - absolutely the victim should not have to move while the bully doesn't!) which is something that they are unlikely to want to do - but it might mean that you will get them to ensure that they are kept on a different table on different sides of the room and at carpet time they are kept away from each other, which would be a good start for your dd.