But would Charles be able to come out as a Catholic here? I think anti-Catholic sentiment is only going to get higher, if the French are pushing to control the Netherlands. Anglo-French relations have never been easy, they're traditional enemies and if the Netherlands are no longer able to act as a buffer state... it could get even worse, I think. Either George of Denmark is married to Mary instead of Anne as part of the Anglo-Danish alliance against a French-Dutch power on the other side of the North Sea, or if there is real fear of a Catholic Succession, I could see Charles either being forced to set Catherine aside in favour of a Protestant Princess - if he has time to father a son, I'm not sure of dates in this period, or perhaps Monmouth is even legitimised and Parliament swallow their distaste of him because a Protestant male of dubious birth is better than a Catholic....
And before anyone says its impossible for a legitimised bastard to inherit, it's not. There's precedent for it in the Beauforts. The first act that legitimised them in 1397 said nothing about barring them from the throne. That particular detail wasn't added to the charter until 1403, by their half-brother, Henry IV, who wanted to secure the throne for his own sons...