I've never seen a WI like this before, but I think it's fairly safe to say that Luise was the brains/pants in her marriage. Napoléon referred to her as "my beautiful enemy" and one of his ministers called her "the Prussian harpy". It was her doing that got an audience with Napoléon at Tilsit, and the soldiers loved her - she was known as the "queen of hearts", the "Prussian Madonna" and numerous other titles.
So, what if at some point during the Napoleonic Wars, Friedrich Wilhelm III dies (maybe he and Luise's brother-in-law, Prince Friedrich Ludwig, are killed at the same time) and leaves an underage heir with Luise as regent? Since we know what the personality of Friedrich Wilhelm IV was like, it seems safe to assume that she'd continue to hold influence - even after the regency is officially over. (Note, this assumes that she survives whatever killed her in 1808 and lives to say the 1840s, as well as that Napoleon still rises and falls as per OTL). How does Prussia fare under her aegis? More liberal than OTL? What sort of gains would/could it make at Vienna with her in charge? What impact might her surviving have on her kids (her death profoundly affected the future Wilhelm I and the future Empress of Russia AFAIK)?
I think that this could maybe make for a fun thought exercise.