The problem with this idea, is that there are always going to be those in the English public who are going to be whispering that the queen's conversion isn't sincere. Although, I read in Flora Fraser's bio of Caroline of Brunswick that [Protestant] princesses were generally raised "without religion" so that they could more easily conform to the religion of their husbands. Now this doesn't mean we're breeding agnostics or anything, just that they get taught about God and the Bible, without the dogma/fundamental rules. Their Catholic brethren, OTOH, seemed to have religious instruction as part of their education. Some were only given that (see Louis XV's younger daughters), and little else.
To get a Catholic girl to convert would require some hoops to jump through, José of Portugal might have been light with the papacy, but the OTL marriage offer from Portugal to England was explicitly not to George III, but rather either the duke of Cumberland (who was also spoken of as marrying a princess of Savoy or Modena) IIRC. Now, how do we assure the public that Princess Maria (seems a reasonable name for a Catholic) is not backslidden from the CoE to Rome? IDK. It's gonna be supremely difficult, since there are going to be those that are going to see secret masses and priests in disguise where there aren't any.
A far simpler way of getting it done is by a king - already crowned and annointed - marrying a Catholic. There'll still be whispers, but the various Acts prohibiting marriage to a Catholic are worded in such a way that it speaks of any person of the royal family marrying a papist, rather than the reigning king himself. It's a small loophole, but I would imagine that parliament/England can't do much about it if George II/IV (who both outlived their first/second wives), once crowned, and ruling for x-period, decides to marry a princess of a Catholic country. Of course, the country she hails from is going to play a role in her reception. If it's France/Spain chances are good it won't go over well. If it's some little pinchbeck dukedom in Germany that just happens to be Catholic, it might not even raise a murmur beyond "but she's Catholic". If she's domestic (a Howard, for instance), it won't go over well at all (since what does she bring to the table? The king is thinking with his little brain?), but they'll murmur better an English Catholic than a foreigner, I guess.
PS. Please note that this is simply my interpretation. I could be wrong about a lot of things.