I'm not suggesting you have to accept 50/50. Very few kids do get to enjoy a 50/50 arrangement because very few parents are amicable enough to make it work. My own stepdaughter is in a 50/50 arrangement and the only reason it works for her is because her parents have close to zero contact, which forces amicability, even if it's against one of the parent's wishes.
All I am suggesting is that if you've never stopped the father from seeing the child (and I won't lie, leaving the father off the birth certificate probably won't support this argument in Court), you should not start now, and if you abscond overseas, that is exactly what you'll be doing.
If all you have said is true, namely that you are in no way responsible for the child not seeing the father, then it's unlikely the father will get a 50/50 outcome from Court anyway. Simply put, it wouldn't be in the child's best interests to go from hardly seeing her father at all to suddenly living with him - and away from you - for a week at a time. Thus, alternate weekends is very much a reasonable offer.
In regards to the decisions about education, religion and travel, the only way he would be able to make those decisions without you is by having a Court order, but mediation is the best opportunity to discuss those issues and their outcomes. If he refuses to negotiate, then you can refuse to sign any agreement and things can go back to the way they were. He could even try his hand at getting sole parental responsibility through Court, but the likelihood of success if close to nothing.
The issue with money and financial contribution really should not be given the weight that it currently is, particularly as it was your choice to exclude the father from the birth certificate, thereby forgoing your legal entitlement to child support. As for single parent pension, whether he claims that or not is his business, rather than yours, because it's the government that funds it, not you.
The other thing is, it doesn't matter what your respective financial contributions have been. If you and he can't agree on when he should see the child and he decides to pursue parenting orders, the Court most definitely will grant the child some time with her father, because that's her legal right. It won't have anything to do with how much money he has offered to support the child over the years, and not being on the birth certificate won't exclude him from pursuing parenting orders, either - if no other presumption applies, the Court can order DNA tests and make a declaration of parentage that he is the father, which will put that hurdle to bed in an instant.
One thing is a definite though - if you abscond overseas, you'll be showing the Court you are unable to support and encourage or even facilitate the child's relationship with her father, which runs the added risk of losing custody all together.
So, you really do need to weigh up your options here. Do you agree to some time between child and father, put his name on the birth certificate, claim child support and reduce the likelihood of ending up in Court? Or do you leave the country and risk proceedings where you lose control of the outcome all together?
