Just thinking out loud here, but I've since been thinking that perhaps a girl/girls would somehow end up in a convent/Stuart mesdames given that there weren't even enough princes for the princesses whose fathers ACTUALLY were ass-warming thrones - Portugal, France and Savoy.
And also, I think these girls might end up the same as James II's kids with Mary of Modena, "not important enough but too important". Plus, does anyone know what the financial situation of the Stuarts in Rome was? And obviously a landless princess is acceptable if she's bringing a large enough dowry, but would these girls even be able to afford that?
I believe the popes were supplying a large degree of the Stuarts finances (and palazzo). I think Clementina had some kind of inheritance and of course the Williamite/Anne/Hanoverian govt. never returned Mary of Modena's dowry. Perhaps a Stuart princess would be relevant to her Sobieski heritage? Perhaps to help a prince of Saxony or a Russian client prince eyeing a decreasing Polish throne give some prestige?
If it's a boy in 1729, I don't think that changes anything. The Jacobite chances died at Culloden so the boy (call him James) wouldn't be able to do anything. But he WOULD be able to keep the Stuart claim going, perhaps entering the service of Austria or Russia, or moving to Modena, and not entering the clergy (like Henry) or becoming a dissolute embarrassment (like BPC). Perhaps he could even be used himself a potential candidate for Polish king?