I admire you for giving serious consideration to all possibilities before embarking on a co-parenting partnership. If your parenting partner will have legal status as a parent, he will have equal rights for custody and visitation. Generally, a right to custody includes decisions as to where the child lives, health issues, educational and religious choices.
When you ask if your parenting partner could surrender his right to custody, do you mean the right to have the child live with him or do you mean the full extent of all decisions about child rearing?
A court determines custody based on the best interest of the child. A court will not be bound by an agreement that is not in the best interest of the child.
You need to consider what decisions about child rearing, if any, that you are willing to share with the child’s father.
Then, you and the father can construct an agreement which reflects those choices. Backing up those decisions with facts and a rationale based on the child’s best interests will significantly increase the chance that a court will respect them.
So, the short answer to your question is that a court is unlikely to enforce an agreement that a father will never seek custody. But with a well-documented, fact based agreement, a judge might seriously consider the decisions made by the parents.