This is an unusual scenario. One might hope that the DWP were reliable.
Are you sure that it is paid employment? The DWP is also responsible for paying benefits, some of which are taxable. It may be that the P45 covers your receipt of State Benefit, rather than actual paid employment - which may well have ceased at or around the time that you took up the alternative post.
If you are confident that it definitely IS supposed to be for paid employment - which you never, in fact took up - then I suggest you ask for a Self Assessment Tax Return for the tax year in question, and set out your income(s), etc., as you believe them to be. Use the Additional Information sections of your tax return to highlight why, and where, HMRC will find differences between your figures and those that appear to have been provided to them automatically by employers.
It is quite possible that HMRC will then write 'automatically' saying that they are amending your figures to those they have on record, and you must then appeal - preferably in writing - and (re-)explain why you disagree. HMRC should then open an enquiry.
It is practically quite difficult to prove that you did NOT receive money. One point that may help is that, if the position that you took up (instead of with the DWP) was a full-time role, then you could point out that it would be difficult to have have two paid employments running concurrently on a full-time basis (although this is not of course, impossible - night shifts, etc.). Other than that, it would perhaps be possible to get the DWP to confirm into which bank account they paid "your" money. If the account is nothing to do with you, then that should be evident from the payee details; if it is an account of yours, then provide HMRC with bank statements for the relevant period, which show no money received from the DWP in that period. Of course, it would be much easier if the DWP were simply to say that they had made a mistake.
I hope this is useful.