Good point - but what if, as iOTL, neither Mary nor Anne had any surviving children? Then Anne's heir would still be her TTL Catholic half-sister.
I suppose what'd happen would be an Act of Parliament being rushed through during her short reign.
Not necessarily, after all there are plenty of variables involved.
Now the birth of... Let's call her Princess Charlotte Alfonsina (in honor of her grandfathers, Charles I and Duke Alfonso IV of Modena) would be met with a sigh of relief on the part of the English and Scottish nobility, now James II isn't in the same position that he was in when he agreed to have Mary and Anne raised as Protestants, but he could at least float the idea of letting Charlotte Alfonsina be raised (High Church) Anglican and gamble on trying again with his wife, but their only other child will be Louisa Maria Teresa (as in RL, though her name in this TL might be a bit more English... like Louisa Mary), but James doesn't know that...
In any case if we give James the same lifespan he will die in 1701, outliving Mary, but giving Anne a one year earlier start to her reign than in RL. By the start of Anne's reign in RL she had had all of her failed pregnancies and stillbirths. Princess Charlotte Alfonsina would be 13-14 so there would have been talk of getting her married but at MOST there would only be a betrothal, most likely to a French Catholic Prince, or a Prince of some Catholic family. Anne would tear up such a contract and damn the consequences.
Princess Louisa Mary would be 9 by the way.
Now I could see, at minimum Charlotte Alfonsina being raised as a very High Church Anglican, but under her mother's influence there'd be a very large Catholic influence on the Princesses. But they'd have also been forced to grow up seeing just how many problems their father and mother's faith was causing the family and the nation at large. Under big sister Anne's influence the girls could be brought around to something that the British isles could tolerate.
Now there might be some talk of an Act banning Catholics from getting the thrones, but it might not have the same urgency if Anne's is seen working with her siblings (and obvious heirs) to make them embrace (or at least toe) the Protestant line.
Plus as the Jacobite Wars proved, royal bloodlines have a power on the imagination that mere politicians in Parliament don't possess. Hell a couple of those wars could have been won by the Old Pretender and Bonnie Prince Charles if they had been a bit smarter.
So Anne has a reign that is a year or two longer than in RL and has (at minimum) gotten at least one of her sisters to become an acceptable Protestant heir, ideally that would be at least Princess Charlotte Alfonsina, but in a pinch they could pass that act of Parliament and put the crown on Louisa Mary's head (a 9 year old is easier to re-educate than a 13 year old).
In terms of regnal names... I don't see either Princess reigning under their double-name. Princess Charlotte Alfonsina will probably just reign as Charlotte, Louisa Mary could reign as Mary II.
In RL, the Old Pretender had two sons with his wife, Queen Charlotte could manage a couple sons easily. In RL Louise Maria Teresa died of smallpox at age 19 and was never married, so her fertility is up for debate. But she could bear children as well.