Ok, So you need a product thats all ready been developed, that is is free to use, and one which you can brand/style and pass of as you own!!
I think you should ask your developers to build you one, as i doubt your going to find a solution (decent) that will tick all those boxes.
Anyways, most of the previously suggested solutions have a monthly cost of $20 or so, which is a little over £10 a month, isn't it more cost effective just to by a license on one of the above and or pass the cost onto your client / company?