I'd rather skip the country or move on so she can't trace me than pay her a penny directly or pay the money into a trust fund or pay directly for receipted purchases...
Wow, I really cannot believe that you feel that way. And, if your girlfriend knows you feel that way, she's a real winner too. Way to "man up". Btw, good luck skipping the country bc most countries have extradiction laws that will take your pay and send it to the CHILD anyway.
Also, child support and "access" as you call it, are completely seperate. Clothing, books, etc., if that's your child and you want a relationship with them, are and should be imo, beyond child support. Poland's laws favor the mother, as it rightly should. Because that's more likely to favor the child. And if you didn't sign a birth certificate, no you don't get to choose the name, where the child lives, etc. Plus, if you feel that way that you just said above, you don't sound like a committed loving father to have the right to have say in that sort of thing anyway.
DNA testing should be done first, of course. But other than that, if you are the father, get a lawyer but as a woman, I have no sympathy. Sex every time can lead to pregnancy. They teach you that in middle school. And children deserve child support.
Marriage is temporary but parenting a child bonds two people together. Suck it up and don't sleep with anyone unless you are prepared to be a man and care for the baby for the rest of your life.