In Scotland now, it is increasingly difficult to exclude children.
My LA does not allow permanent exclusions and - to paraphrase Educating Rita - you practically need to bugger the bursar in order to get a short temporary exclusion.
The best that you can do as a school is to make life so awkward that the parent agrees to a whole school transfer in order to move the problem child elsewhere.
As a parent, you need to make life as difficult as possible for the LA.
My reading of the OP is that, while there has been one truly awful assault, this has been preceded by several more minor assaults - the pushing. If these minor assaults had been addressed, the most serious assault might not have happened.
The age of the perpetrator is immaterial so far as the victim and her family are concerned. [Very relevant with regard to how the police would view matters, I agree. Children under the age of 16 here are dealt with by the Children's Panel. The Panel can deal with children who are in danger, who are out of control or both.]
If the school and/or the LA focus on the age of the perpetrator, the victim's parent needs to be assertive and to point out that their concern is to keep the victim safe. It's not good enough for them to say "What can we do...?" They have a duty of care to their pupils and should state in writing what they intend to do in order to keep the victim safe in the future.
Similarly, if the school says "...but she has a terrible home life..." Not. Your. Problem. However, if the school stupidly does this and unwisely tries to downplay the behaviour as a "trauma response" ["stupidly" because this is privileged information] then the victim's parent might point out that the school has a duty to inform the authorities that the bully is out of control.
NB Sometimes a school will go down the road of suggesting a separate playtime for the victim and/or a later start time and earlier dismissal in order to keep them away from the bully.
This is unacceptable. If you have a separate playtime, you are being deprived of interaction with your peers. If you come to school later and leave earlier, you are being deprived of part of your education and of interaction with your peers.
If the school says "But what about the rights of [the bully]?" that is not the problem of the victim's parent.
The victim's parent needs to know what the school [and by extension the LA] are doing to safeguard their child. This might be of interest: https://education.gov.scot/media/uzdfwccr/es-safeguarding-and-child-protection-policy-aug-24.pdf
Just to add that if I were the victim's parent I'd want to check with a dentist as to whether the damage done might impact on permanent teeth as they grow in. Remember to take photographs of the injuries.
Also, is the child currently experiencing difficulty with eating or speaking because of the damage?
If the child is going to need future dental treatment, I'd say that the LA might be liable if their agent - the school - was already aware of the 'minor' assaults and failed to deal with them. No, I'm not a legal expert - but there are plenty of solicitors who offer a free half-hour consultation. (They're ambulance chasers to an extent, but Thomson's would be an example.)
It would be impossible to sue the bully's parents, but the LA is a different matter.
I know of very few cases where parents have resorted to calling on the services of a solicitor, but I can say that S.L.T.s are frightened of the prospect of parents doing so.
I hope that the above makes sense. It's late and my sentences may be less than coherent.
I'm also enraged by the harm done to the OP's daughter. When I was a bit older than her, another child damaged my adult front teeth as they were just coming in. I needed extensive dental treatment as a result.