School Governor here - how has recently had too much experience of this sadly!
Firstly you need to think about where you want your child to be in the future. I absolutely appreciate your child being black and not wanting to be isolated from her culture comes into this. She is Year 8 - so has time before things like GCSE options take hold (For some schools this is in year 9).
If there is a school local to you (within the same local authority) that you would be happy for your child to move to check with the school that they have space - or you could ring the local authority and ask who has spaces in year 8 as this would avoid having to tell the new school that she is PEx currenty. (Schools cannot go over PAN to take in a managed move pupil which will limit your options) Then approach the current head - say that you have reconsidered managed move and would like to discuss futher. They do not have to agree at this point as they have already excluded BUT they could be open to it if they think you are serious (it does get used to get out of exclusions so many are wary). There are timescales at play however now the exclusion has been called, so you will need to act quickly. Managed move does feel like the best option if you can find the right school as it does not feel like the school is properly supporting your child from what you are saying.
If the managed move is not possible then you will go forward with the PEx, and there will be a governor panel within 15 school days of the exclusion- made up of the schools own governors. Prior to this your daughter will receive set work from her school for 5 days, and then from the Local Authority or Inclusion partnership for the remaining time until the panel. Make sure she does and submits this work please. It does make a difference to how governors feel about kids who they are told are bright, want to learn etc.
At the panel you will be asked to present your side- and you can state that your daughter says she did not push the teacher. You can bring evidence from anyone else in the room at the time who might confirm that. You can challenge anyone elses evidence - for example if a teacher who was not in the room said it happened - how does he/she know if they were not there? You can also add in all mitigating circumstances - anything on school records about the incident type - you mention she has had issues with this teacher before for example.
Which way this goes will depend on the governors. If they are convinced your daugther did not push the teacher or that it was a 'shove past' in attempt to get out of the door that the teacher blocked for example they may overturn - but don't get your hopes up on this.
The governors have to be convinced (within the civil standard of proof which is reasonable probablity rather than the criminal one of beyond reasonable doubt) that the action took place.
Having decided it did take place they then have to consider:
Whether the exclusion is proportionate
Whether there are any protected characteristics at play - so your daughter being black will come in here
Whether having your daughter back in school is a risk to the school community
I would be arguing that it is disproportionate - look at the schools behaviour policy for the argument here - though if they class it as violent then it will meet that criteria, If you think it is targeted on her race ask how many black/em students have been excluded by the school in the last 2 years in comparison to non BME, also talk about the impacts of moving a black child into a non BME school. The risk to the community is difficult to disprove or prove tbh, she clearly disrupted a class at the time of the incident, took teacher time to resolve, and returning her will have an impact on the percived safety of others - but again there is a proprotionality to come in here.
Youve been advised above to seek legal advice. I'd look to having someone with you at the meeting - paying a solicior is rarely an advantage as they just railroad and delay the process but likely wont change the result - in my limited experience. But taking advice either from a charity, a IPSEA if there is one in your area or something similar will help. There are also boards on here which might help further.
Good luck OP - put your child at the centre of this - not what you think the school should do and you should get a decent result!