I moved my sons from State to Private 2 years ago. They are 9/10.
I often buy stuff like juice and treats at Lidl.
Oldest boy sometimes takes a carton of Vitafit (lidl brand) to school. He said some of the boys had laughed at him drinking the wrong orange juice. Luckily, his instinct was to laugh too and say :" the WRONG orange juice, what can be wrong about orange juice, you are not making sense". When they mentioned it again he just said "whatever" in a genuinely unconcerned way. He can still giggle about "the wrong orange juice" at times.
That was that. But if they would have touched a nerve, I might have given him unbranded stuff (ie orange juice in drinks bottle). Though I am stubborn enough notbto change my shopping habits for some little brats, and would most likely let DC find a way of dealing with this.
Conformism is just so boring.
I strongly believe you have to help children to become resilient and deal with this kind of bullying. Teaching kids this age the power of "whatever" can be useful.
Incidentally, everyone in our school goes skiing, it is all everyone talks about in winter. People are just astounded that we don't ski. It is not something we could or would afford easily. Step in DH who is scruffy countryside posh, saying " I can't believe anyone would just disappear for a week in the middle of the hunting season!" That shut up the nouveau's.
Anyway, hope you sort it out and your boy settles in. It can be hard at this age. Support him, listen to him, amd help him come up with a strategy to deflect the bullies.
The are lots of people at private school who are not actually super wealthy and scrimp and save in other area's, witness the 2nd hand uniform shop!
So you don't need to posh up to fit in.