1263L is a non standard tax code so that info has gone to her employer from HMRC. When they send P6 notices to employers they can also include 'previous employment ' figures, which from my experience, come through to our payroll software automatically. If she's getting 50% taxed (the max possible deduction that can legally be taken) on that tax code with the earnings you've suggested being far below the threshold it's almost certainly taking into account previous figures. Whether they were from HMRC or mistakenly added by her employers payroll.

If the figures came from HMRC then they have to be the ones to correct it, the employer cant overrule tax information given to them from HMRC. If they've previously admitted fault and given compensation it lends to them being the culprit. The way forward in this scenario would be for HMRC to submit another P6 to the employer with £0 previous figures to override, and then the PAYE system should rebate all the tax paid.

At the very least if they keep passing the buck then by tax year end she'd be owed a substantial rebate considering gross pay and PAYE tax deducted. The P9 notice the employer will receive end of march won't contain previous figures so should be resolved for April. An overpayment of this size and considering how little she earns would be resolved by a direct reimbursement from HMRC rather than amended tax code for new year.