oh gulp!
Yes in that case you are right to go to the head.
My DS has had similar trouble going back a few years but not neighbours, not working at school. Sorry OP - didn't realise you'd gone to class teacher(s) and got nowhere.
Because it's quite a developed thing now and you've already been to see class teacher on several occasions, I hope you don't mind me suggesting these things that helped my sister when she went to her DS's HT.
Initially she went in with a - I know that I sound like an over protective mother but ... attitude and basically spelt out the problem ... then waited and nothing happened.
She then went back with a this is still happening, can you please tell me what strategies you have been using to deal with this behaviour because they don't seem to be working? attitude which worked much better.
She could have saved a lot of time and her DS some heartache if she'd used the most effective attitude first!
Maybe you might need to point out that it's not acceptable to simply not notice this behaviour - 5 year olds can be sneaky and are not stupid, of course she won't be doing in ear shot of the teachers or under their noses! She'll be choosing her moments carefully when she can get away with it!
Support staff, class teacher, dinner ladies all need to be aware of it and be discreetly but actively looking for it so that it can be dealt with.
My sister also said when she went in the 2nd time I'll leave it a couple of weeks then if it's not resolved I'll come back and make another appointment.
Hopefully though the head will sort it out promptly - but you can use that one if you feel you're not getting anywhere.
In the past class teachers have fobbed me off with "I'll do a whole class talk about being friends and kindness" which I've thought was a waste of time but did work, but it sounds like your DD's situation is far beyond this and that it's very calculated.
You can also tell the HT that you've been trying role play, told your DD to ignore it, to challenge it ... bascially everything you can think of to help but that it is not enough and that for your daughter's well being and education you need for the school to be playing an active part in solving this issue.
Good luck OP and good luck your daughter 