Which exam board is this with? I assume the £350 is for
three A levels, is that correct? It's worth checking that the fee you're being charged (excluding the admin fee, where the school can charge what they like) is accurate. For example, AQA make it clear
here that the "late entry fee" is "100% of the original entry fee and is in addition to the entry fee." OCR same the same
here: "Late entry (Stage 1): Charged for each entry received after the entry deadline. This fee is the equivalent of the original entry fee and is charged in addition to the entry fee". So for both of these (and I
assume the rest, but haven't checked) the £350 should have doubled to £700. Where's the extra £50 going? To the school?!
If you have evidence that you provided details and payment two weeks ago, and if this is prior to any
internal deadline set by the school (which is likely to be ahead of the exam board's 21st February deadline), then remind the exam officer that the schools had a
legal obligation to fulfil the contract you and and it entered into. (I assume you have the appropriate paperwork to back this up.)
I would be tempted to point out that the school would be at risk of losing its accreditation with the exam board if you were to able to successfully prove to the exam board that the school had missed the deadline through incompetence, or had done so intentionally in order to profit from doing so. There's obviously a risk, however, that this approach might backfire.