I tried to do a TL on topic - I was told that once the heir to Margot and Henri is born, and it's clear that he's healthy, Henri's mother-in-law will try to organize his assassination and claim the rights to the grandson's upbringing.
I'm sure there are people here who know the details of the period better than I do but it just seems like that wouldn't have to be the case. I understand Catherine's reputation but if the plan was just to assassinate protestants to move a Catholic up the line of succession that could have been done easily OTL. Really when you look at the line of succession in say 1584 at the death of the Duc d'Anjou its mostly Catholics. Granted some of the Conde brothers have Protestant sympathies but I think that's mostly due to family connections and they could be expected to rule as (tolerant) Catholics. So if the opportunity was there OTL and wasn't used I don't see why we should automatically assume it would happen in an ATL where Henry has a son with Margaret.
heir apparent in 1584:
1. Henry de Bourbon, King of Navarre
followed by;
2. Cardinal Charles de Bourbon, Catholic, his uncle, unmarried
3. Henri de Bourbon, Prince of Conde, Protestant, his 1st cousin, widower, no heir
4. Francois de Bourbon, Prince of Conti, Catholic, his 1st cousin, married, no heir
5. Cardinal Charles de Bourbon, Catholic, his 1st cousin, unmarried
6. Charles de Bourbon, Count of Soissons, Catholic, his 1st cousin, unmarried
7. Francois de Bourbon, Duke of Montpensier, Catholic, his 2nd cousin once removed, married with heir
8. Henri de Bourbon-Montpensier, Catholic, his 3rd cousin, a child of eleven
I suppose you could argue that a Prince of Navarre might ensure a protestant succession, assuming he's raised protestant, and that might threaten Catholic interests more than the OTL situation, given that several of the Catholics on the list, as Cardinals, can't have children. Though, on the other hand, the only protestant, Henri of Bourbon-Conde, is a widower without an heir while Soissons and Henri de Montpensier are young enough to be expected to marry and have children so on balance it sill favors a lot of Catholic lines. Still, this maybe forces the ultimate solution that the young prince is raised Catholic while Henry himself is not forced to convert. Then maybe the Catholic faction would tolerate him secure in the knowledge that the succession would ultimately go to a Catholic. It might be a bit of a stretch considering the intolerance on both sides. But I have to assume that long term someone will have to convert because a protestant, or at least a protestant dynasty, is simply untenable on the throne of France at that point in time.
Of course, no matter what, the child would escape the clutches of Queen Catherine. If that's the point that was being made, that she'd want another child King she could manipulate, then I suppose I could see it but it still seems like it would be hard to pull off. It would clearly be a personal act and there are other Catholics that could step in to raise the boy who would be less objectionable to the Bourbons. Why not place the boy in the care of his Cardinal-cousin along with his Catholic mother. I'm sure the whole Bourbon dynasty would rally around him and try to shield him from Catherine and/or the Guises.