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The (Napoléonic) Bachelorette, Season 4: Who Will Luise Pick?
Let's look for bridegrooms:
So, the duke of Modena was married by 1812, which makes him an unlikely candidate for Maria Luise to remarry to.
Kings of Naples and Sardinia were both married.
The grand duke of Tuscany was a widower who only remarried in 1821. (but he's also Luise's uncle, so while the Habsburgs did this type of thing, I don't think Franz II would be interested in having his grandkids only be able to inherit if his nephew dies.
The duke of Parma might work - albeit he's eight years younger than Luise. Which basically rules out Italy.
Fernando VII of Spain only remarried in 1816, so Franz II might think a double Iberian suit (Luise to Fernando and Leopoldina to Pedro) might do nicely. His second wife died in 1818, if the annulment's not ready by then (although OTL Luise has had three of her four kids by 1818).
The Bourbons in France wouldn't be interested - although for sheer hilarity, I would pay to see Franz II annul Luise's marriage to Napoléon and marry her to the duc de Berri. The Bourbon line was running low on boys, they needed a match sooner rather than later, with a girl who was known to be fertile.
Ludwig I of Bavaria's married since 1810, but his only brother married morganatically in 1823.
The future king of Saxony is too young for Luise (besides, Saxony was on the losing side, you don't still want to reward them)