i feel this is like pulling teeth, Kellan. But here goes.
Bourgogne marries Antoinette, Berri marries one of the two brides originally proposed OTL Luisa Maria of Parma (yes, the OTL queen of Spain) or Maria Amalie of Saxony-Poland (otl countess palatine of Zweibrucken).
That done, either Luisa rides rougbshod over Berri as she did Carlos IV, and provides an extensive brood of Berri children (as she did in Spain, paternity not necessarily being her husbands). Or Amalie comes in and bears her husband NO children, which I think is more likely
Either way Berri is probably not going to be seen as an overly ambitious character and not going to be regarded as a threat to the king.
Though what I want to know is what Bourgogne would do when the Revolution breaks out. Otl Louis XVI sank into depression as result of the Dauphin's death and later because of his refusal to accede to even the smallest increment of his power being removed, the convention went ahead without him, testifying to Napoleon III's phrase of "march with the ideas of yyour era or be dragged along behind them". With Bourgognes stronger personality I could see him being either a) more pragmatic; or b) more devious; either way carving out a niche for the king by exploiting the factionalism in French politics. He might even divide the royal family a la Napoleon III into a famille royale (king, queen, dauphin and their other children, Monsieur, Madame, their children) and a famille civile (Provence, Artois, Orleans, Conde and Conti and their families [Provence and Artois I'm not sure which they'll actually fall into])