The only thing I can offer by way of experience with this, is opening a savings account as a non resident parent. There were various schemes such as child trust funds (they stopped those) and possible child ISA's that I think were only available to a resident parent. I tried opening a child savings account with a building society (they needed all sorts of ID for the child - copy of birth certificate and passport etc) but gave up trying to get that one - can't remember why. In the end it was Barclays bank that allowed any adult to set up a savings account for a child. It was basically shown up on my own online banking as a child savings account so although it was in my names it was designated as a child savings account for xxxxxx child. That wasn't the best interest of course but it was something, and it did grow over the years putting £10 a month in.
Halifax has this one with 3.5% interest - for an adult to open for a child. It just says if you aren't a parent or guardian you need a parent's permission. It doesn't say anything about parental responsibility though. However, if you're not on the child's birth certificate, it could be difficult to prove you're a parent.
One option could be to get a one off solicitor's letter "to whom it may concern" stating that you are the parent of xxxxxxx despite not being on the child's birth certificate.
Halifax kids saver account
My question is - have you tried to get a parental responsibility order? That might make things easier!