I have got something like that I suppose from a youth group I used to help with. As I wasn't an employee of the youth group... it is I suppose a 'character reference' rather than an employment reference.
Why are they wanting a non-childcare reference? Surely references from families you have babysat for, will be better than one from a friend.
Choose a different agency! An agency won't want a character reference, they will want childcare references... something that says how great you are as a person caring for children, plus your reliability etc.
Are they wanting such a reference due to a lack of childcare ones perhaps?
What have you supplied them so far? Your current employer has written a reference I expect. Prior to your current job, have you got references from past work (be they childcare or non-childcare jobs)? Have you done any Youth Group volunteering... group leaders are often willing to write about they found your help invaluable, how you were reliable, trustworthy etc. Get families you babysit for to write a short reference (a couple of paragraphs is fine I find).
Seems enough to me. A good agency will want phone numbers of people to contact, so they can write up their own 'reference' as it were. You are providing 4 written references, of which at least two I expect can be followed up quite easily, possibly all 4 can be followed up.
Try different agencies... what you have to me seems to be sufficient to get you on their books, given they are not making an NVQ3 in Childcare a criteria.
If you can meet the requirements - being Ofsted Registered can help. But your Health & Social Care certificate may not be sufficient.
You seem to have it all in place... so just try other agencies and look privately. The best agencies I find are those run by Recruitment Consultants, not by ex-nannies.
Try to add more childcare references, you can never have enough. Families you babysit, for example.
Make your CV specific to each job you apply for... don't use a generic CV for all jobs. Try to make it emphasise the qualities that the employer is looking for. Don't just list, I worked at x place... instead write a paragraph or so under each RELEVANT job detailing what your duties were, what you did, etc.
I've had this and found there's nothing you can do to change the attitude that unless you have 10 years sole charge experience you are not worth the time of day. It's a real shame. Keep looking privately and nannynick has good advice (as always) Have you tried advertising yourself (get a lot of time wasters but worth a try) Just out of interest do you find agencies accept references from any position that isn't strictly a nanny because they were all really shirty with me over that? Good luck