Of course, it should be noted that the Prince of Wales's union with Mrs. Fitzherbert WAS a known entity to George III to the point of the latter actually pensioning the woman even if he refused to acknowledge her as his daughter-in-law [and it seems he and Queen Charlotte actually liked her as a person]. George III simply waited out Prinny's spendthrift ways getting him into such deep debt that even his royal connections and being a heartbeat from the throne weren't enough to keep the creditors at bay. Hence George III made a bargain to pay off his heir's debts in exchange for Prinny marrying the Suitable Protestant Princess bride of George III's choice[his own niece] whilst everyone pretended the marriage between Prinny and Mrs. Fitzherbert was invalid or had never happened. It's interesting that even though the Prince of Wales had been notoriously unfaithful to Mrs. Fitzherbert and had already moved onto to more permanent mistresses even before the Brunswick Stew, he made several declarations that he considered Mrs. Fitzherbert to be the wife of his heart AND he wore her miniature on his neck and insisted on being buried with it after he became king.
If the Prince of Wales had somehow learned to even try to live within his means, he may have been willing to risk losing his place of succession for Mrs. Fitzherbert even if he wasn't willing to be faithful to her.
Actually the Prince of Wales is the one who suggested Caroline of Brunswick, not George III. From what I've read in
Princesses by Flora Fraser, the Prince's decision to get married seems to have taken most of the family by surprise. As for losing his place in succession, I doubt it. The Prince loved the trappings of Royalty and Monarchy to much to give up the throne over a woman.
Plus, according the the Royal Marriages act, he was never married. Without the Sovereign's consent or the de facto approval of Parliament such a marriage would have never existed in legal terms. At most, if George never married a royal, we would probably see something similar to Louis XIV and Madame de Maintenon, Elizabeth of Russia and Alexey Razumovsk (and possibly Catherine the Great and Grigory Potemkin, depending on the source): an unacknowledged but well known marriage. Maybe if they were still together when he ascended Mrs. Fitzherbert would be made a Duchess and could possibly influence George IV in favor of Catholic emancipation.
Really though, publicly acknowledging the marriage between Wales and Fitzherbert would end up causing a big scandal that would damage the prestige of the Monarchy, the Government and George III himself. Neither the Whigs or the Tories would really win anything by doing so. So it would never happen.