With a son (even a Protestant one) it is less likely that James II & VII would feel a need to remarry. No warming pan baby, perhaps no Glorious Revolution (though there was more than one thread in that warp) , a very different War of Spanish Succession.
So much may hang on a single life.
Charles would never agree to his brother's exclusion. Not just out of filial affection, but because he knew that the exclusion demands were a lever to try to secure an elective or nominative monarchy.
I dunno, the Duke of York was already drifting towards Catholicism. Anne Hyde converted shortly after the Restoration, actually. James was the one who followed suit nine years later, although he continued to attend Anglican services into the mid-1670s. There was a reason why Charles II insisted his daughters Mary and Anne were raised as Protestants. He'd insist the same for his son.
Assuming James II still converts, though, it does have a plus side. Having a Protestant heir means the Glorious Revolution is pretty much out of the picture. An eldest son cannot be displaced by any other, so the Protestant party in England would merely view James II's reign as a bizarre interragnum, rather than something to be feared. This would be doubly true if James III is an active supporter of the pre-cursor to the Whigs. Having his son as a Protestant might still greatly aid James' image as he would probably be much less radical in his thinking given that the Protestant succession is ensured. He would probably still remarry though, even if James III lives. One male child isn't enough; with Charles II and Catherine of Braganza having no issue, having both an heir and a spare is quite important, especially in the age where children often died young.
One plus might be earlier religious tolerance. If James II reigns and doesn't mess things up too badly, his son might be keen in mimicking his uncle by issuing his own
Royal Declaration of Indulgence, probably by abolishing the Test Acts and suspending other penal laws directed against recusants, thereby granting both Catholics and non-Conformists religious liberty. If his father's reign isn't too bad, Parliament would be much more willing to accept such an act, although during Charles II's reign, it was the Cavalier Parliament who came down against him, so you never know.
James III though, will definitely have a Catholic princess in the cards as a potential bride, even if raised Protestant. Both Mary and Anne eventually married Protestants, but Mary herself was briefly considered as a potential match for
Le Grand Dauphin (what a mess that would be... gives me an idea for a WI of my own, actually!) by her uncle, so I could see Charles thinking that way for his nephew; if Mary and Anne both marry their OTL husbands, then the Duke of York would be quite adamant he marry a Catholic princess, such as
Anna Luisa de' Medici, or any Italian or German state. A Protestant match is quite possible too, though, it merely depends. But with a Protestant heir I think the populace of Britain will be much more content assuming he himself is not influenced by his father in any direction that sways him towards Catholicism.