What you are missing is that one way it could be interpreted is that they want to get their £1260.35 GROSS pay back. There is no need to get HMRC involved, or pension, or student loans. You don't need to get any money from them. Your employer just needs to give them less money in future, just like they need to give you less money in future.
All that needs to happen is for them to reduce your future gross pay by £1260.35. What they want to discuss is a suitable schedule, instead of just docking £1260.35 from your next pay slip (maybe you spent the advanced monies and live from one paycheque to the next).
Effectively what you've had is an advance in future pay. You are not "out of pocket", quite the opposite in fact - you are "in pocket" because you've been given money you haven't earned yet now the employer just needs to offset your future earnings to correct it. Because it doesn't span a tax year, it should make things easy.
The key thing to clarify them is what they mean by "repayment" - my assumption is that any competent payroll department would proceed as I have suggested above. It's arguably ambiguous terminology as really the 'repayment' isn't you paying anyone any money, it is them paying you less money.