Well, you will need the birth cert to get the exact date of birth.
So, you can either go round in circles, or bite the bullet and get that Halstead birth cert. All in all it looks very likely. If the birth date ties up, then you will have her parents names. If it doesn't, then that obvious one will be ruled out. But unless that one is ruled out, then you will keep coming back to it as being the most likely, but will never know whether to look elsewhere or not!
One cert - and it could all be sorted!