... .an offer to her attorney a. She has full custody and at her discretion on IF and when I get visits. My discretion on financial support. Keep my name on birth certificate. Or B. One weekend per month w financial support. Or C. Battle it out in court.
Why only one weekend per month? One typical perspective with new members is that most are willing to sell themselves short. If you want to stand up for the child, then stand up. Standard is alternate weekends or else 2-3 weekends per month (such as first, third and fifth weekends).
Repeat, if you will be getting visitation, then know full well that all courts consider alternate weekends (mine were long, full 3 day weekends, Fri pm to Mon pm, or Friday after school to Monday before school) is the least a parent gets unless there is some level of concern about that parent's parenting.
Also, courts are very unwilling to order sole custody to one parent unless there is solid reason to do so. Or if the parent wants to give up custodial responsibility.
Also, I don't think a father can avoid child support. My court said all mother had to do was to file for it and they'd pull out the calculator. So even if she agrees not to file for CS and it becomes part of the order, she could later seek CS and probably they'd allow it. Of course, if you're not viewed by the court as a parent they may not come after you later. But get a legal opinion about that too. Remember the old saying, the worst patient is the person who has himself for a doctor? Same goes for legal advice, you're not a lawyer so don't try to be your own lawyer. Even our excellent peer support here has its limits, we have tons of hard won experience but we still can't give legal advice and especially not local legal advice.
Also, understand that when you present an offer, you may view it as this or that. Ex will view it as your starting point and she'll be focused on whittling it down smaller and smaller. I recall I started out being 'fair'. I told my then-stbEx I thought 50% time was right. She totally refused (she demanded that I have supervised visits!) but with the poor reports eventually before the court at literally the last minute, just before the long awaited Trial was to start, she settled and I not only walked out with equal time but also as Primary parent. While fathers have an uphill struggle to get results that good, the point is that you don't have to settle for crumbs. Frankly, you might end up with crumbs if both she and court are unhelpful but it doesn't mean you should start out asking for crumbs!

Also, any offer you make can have a deadline attached. An offer is not a forever one, it can expire or even be rescinded. Nothing is "written in stone" until the settlement is made a court order. Even then, it can be modified after a year or two and especially if there are major changes to either parent's circumstances.