I'm sorry to hear that you are in such an unpleasant dilemma. Your child being bullied at schol and not receiving the right kind of action from the school is extremely unpleasant. It's not easy dealing with this.
I'm sure you have done all the below steps I'm about to write but just in case I'll write anyway: I'm sure you have explored the usual routes ie speaking to the teacher, then KS2 leader, and if they were unsatisfactory then the next stage is the headmaster, then the governing body.
There must be a specific governor in charge of child protection and anti bullying, someone DSL (Designated Safeguarding Lead) and you should write to him/her as well as the Chair of Governors.
Please always put everything in writing and create a trail of accountability when you deal with anyone in the school with regards to bullying. After speaking to the teacher, headmaster ..... to whoever, always re-cap what has being said with a letter / email to school.
Obtain the copies of safeguarding/anti bullying/behaviour and complaints policies as well as school's Customer Services policy which outlines how fast complaints will be dealt with, how are they to be dealt with etc. These will be your point of reference.
In some boroughs there are Parent Partnership Service and you can ring them and ask for advice. I heard that they can accompany you to the meetings with the school and it's a good idea to have that extra support in a meeting with the school. Someone on your side and betterstill, familiar with dealing with schools. Having said that, unfortunately not all boroughs have Parent Partnership Service.
Schools are hot on attendance and they are under pressure to achieve certain attendance level. When you talk to the school mention that your child does not want to come to school as a result of the unresolved bullying incident, this will mean compromise in their attendance levels. It should (in theory) alert them but if a school is inefficient in dealing with bullying probably will be inefficient in being proactive with attendance too. However it's worth a try I believe.
If you cannot get the current school sort out the bullying, you should leave an appropriate feedback in Parentview, the Ofsted website about the school.
Local MP might be another avenue to explore if your dealings with the school doesn't resolve the matter, arranging a letter from the local MP to the school, for instance asking for school to follow the correct procedure and policies and asking them to do the right thing, gets logged in your child's file, raises your child's profile which is a good thing.
If all these are unsatisfactory then changing schools would be an option. You mention the alternative school has a bad reputation for bullying, is it on school's Parentview site? Or is it via talking to other parents whose children are at school? How reliable is it? Have you been to the alternative school, talked to the headmaster? How is your gut feeling after visiting the alternative school and speaking to the headmaster?
It sounds like home education is a viable option for you. After exploring the above options you can always opt for home ed.
I would also recommend getting your child to some assertiveness workshop. I highly recommend ZAP
anti bullying workshop for your child :
www.kidscape.org.uk/projects/zap-anti-bullying-and-assertiveness-workshops/
Best of luck.