You may have a bit of a chicken and egg situation in that you would need to apply for a visa before you can come to France; if you could prove family ties the visa would be issued automatically subject to you meeting a few conditions, but since you can’t even start trying to prove family ties until you get here, you would need to find another route to obtaining a visa. You say you “wouldn’t be opposed to learning French and moving to Paris” but US citizens are subject to immigration conditions, they can’t just decide they will maybe move to France.
I can’t see any court deciding that it would be in a child’s best interests to be shipped half way around the world on a regular basis. Think of what impact that would have on its education, its social development etc? Worst case scenario, even suggesting this might go against you because the judge may feel you are putting your own wishes - to get shared custody of the child - ahead of the child’s interests.
Privately arranged paternity tests are illegal in France. A paternity test can only be instructed by a judge in the context of formal proceedings brought by either the mother or the putative father.
If you want to pursue this you are going to have to think hard about how to present your case I think, because looked at objectively, the way you presented it in your first post does not show you in a good light. You have a fling with your ex (what were the circumstances of the split?), who you don’t trust and you think has mental issues (and yet you went ahead and had a fling with her?) and is incapable of raising a child, and you get her pregnant (you didn’t use contraception?). Then you set a private detective on her. Is a judge going to warm to you and see you as a reliable, supportive parent and role model?
Be aware, too, that if you do succeed in establishing paternity, you will have ongoing responsibilities and obligations and France will make sure you meet them. French law is big on making sure that rights are balanced out by responsibilities; you won’t be allowed to dip in and out of being a parent. If the matter goes to court, which seems likely since from what you say the mother is likely to contest your claim, and if the courts decide to grant you any rights, the they will also decide exactly what obligations you will have in return for those rights. Be careful what you wish for, as they say.