First things first the SLC (student loans company- the national brand that the vast majority of people use, if you went with a bank or something then chances are it is a proper loan) are, contrary to the image they try to put out with the we are not an ordinary loan stuff, much like any other debt/insurance/finance/legal type and as such they will try every trick in the book to try and make you admit things or lead you down the path most beneficial for them. Always be on guard when dealing with them and read the smallprint.
Anyhow they did the same for me when I was not claiming benefits (I later did when it seemed that their money would pay for an OU course and parted when I had enough of them being absolutely useless and actually doing more to harm my chances of getting somewhere than helping- a rant for a different day though).
At the time I merely wrote a letter myself and had the people I was staying with (in this case your parents) write a letter to them (they should have am address)- if you were lying such a letter would seriously help them in court so it is good enough for proof in many cases.
You should just be able to point to your national insurance "statement"/contributions.
If you do want to sign on there is a phone number that is probably easier than finding yourself at one of their shops in a town somewhere- most of it is name, DOB, when you last worked and address as well as NI number). If nothing else when you leave them they will afford you a nice P45 which helps a bit during interviews- the amount of paperwork to transfer someone is easy enough and reasonably common but a new employee is less fun.
The easier way if it can so be called and if they will not accept other proof (the most likely will) is to call yourself self employed (you will* need to register a company, ultimately file tax returns/keep books and you will probably want to file for a "certificate of small earnings exception" for National insurance too which is form CF10-
http://www.hmrc.gov.uk/selfemployed/fagcf10.shtml )- there are payment thresholds below which you can carry on as you are doing (not to mention have proof you are below payment threshold for the SLC) and in doing so it does help if you want to pick up extra jobs here and there (contractor/freelance style- much to my dismay we are witnessing the slow death of cash in hand jobs meaning having a legit option is good) or indeed make your own work not to mention far better CV padding than what you have now.
I would advise that if you are not so familiar with these sorts of things and do not fancy reading up (or indeed want to take it seriously and get some of the breaks/benefits/grants) then have a chat with an accountant (you are a prospective client and they are trying to sell themselves so charging you will probably not happen).
*all of this can be done over a phone/internet or with free software. On the maybe side of things a business bank account might help things if you use your present account for non business things.
Also +1 to Da Mafia, the inland revenue have been known to sting people for such things.