Hi L.,
First, I'll answer that I'm Catholic and have been all of my life. Both of my kids have been baptized as infants. If the kids have not been adopted by you, the biological parents, if listed on the birth certificate, must be on the baptism certificate. The baptismal records, along with the other sacraments, are legal documents for the Catholic church and will be used for the rest of their lives. It also provides back up documentation in case of needing to prove identity (for example, my husband's family lost his birth certificate and the wonderful state of LA couldn't "find his" in their records, so to get a passport, he had to use his baptismal record, along with other random documents to prove his name, age, etc - also, we weren't married or even together at the time our first daughter was born - my now husband, her bio dad, is not Catholic and it didn't make sense to me that he would need to be on the baptism certificate, but he is due to their policy).
Second, you need to ask for clarification with the church regarding their "yearly fee". Many churches do push for people to be active members in their parish before undertaking a sacrament. Part of active membership can be defined as giving back to your parish, in the form of time/talent and/or monetary donations. If you do not agree with their policies or do not feel comfortable there, please look to a different Catholic church. Each parish has its own policies and culture, so I'm sure you can find one you feel at home at.
On a different note, I think you should ask yourself why you want her to be baptized, especially in the Catholic church (you don't need to answer to any of us). Different religions view the process and reasons behind baptism differently. It is not up to any of us answering on this post to tell you the "right" way to baptize (and on that note, in my parish, 5 year olds are fully submerged under water in the baptismal font with the priest or deacon - not "sprinkled" with water as others suggested). If you have beliefs that are in line with one religion over another, that should help direct you. The Catholic church has parent classes prior to baptism so they can understand the Catholic theology and what baptism really means as that child grows in the Catholic church.
I will speak my opinion on this situation now - obviously, I do not know the background or reasons why the parents haven't been involved and now are. That being said, I believe that in situations when possible, religious matters are best to be discussed among those who directly impact the child's upbringing. Obviously the parents are involved (to what degree, I don't know) and while you have custody, you have not been granted to adopt her. So, I think that leaves the possibility (again, not sure to what extent) that her biological parents could be back in her life more and more. I think it'd be very hard and religiously detrimental on a young child to be baptized at that age and then have a situation where they have adults that are unsupportive or uneducated of that pathway as they grow.
I wish you the best of luck as you move forward in this situation. I'm sure it's tricky and emotional to deal with.
~L.