What if Joséphine had at least one healthy son with Napoleon? How might it change things?
A son of Napoleon and Josephine will change everything as Josephine would stay as Napoleon’s wife and Empress and Napoleon's efforts in 1808/10 would be on finding Royal brides for both Eugene and Napoleon II (Maria Amalia of Naples/Augusta of Bavaria for the first, Anna of Russia/Elisabeth Ludovika or Amalia of Bavaria for the second).
Plus a son by Napoleon and Josephine would butterfly without doubt the wedding between Hortense and Louis. Hortense will be then married to one of Napoleon’s most trusted officers while Louis will be available for marrying the Dowager Queen of Etruria instead of Lucien (and that would likely reduce the rift between Lucien and Napoleon and maybe made the second wedding of Lucien more tolerable to Napoleon and Alexandrine accepted by him as member of the family).
If Hortense became widowed early she can be offered as wife for Ferdinand VII of Spain (instead of the OTL half-promise of Lucien’s eldest daughter) as she would be ready for re-marrying in 1808 (being still in her middle twenties) unlike Napoleon’s eldest nieces as Lucien’s eldest daughter Christine was only 13 years old and a surviving Julie (Joseph’s eldest daughter) would be only 12 years old.
In alternative Hortense can be remarried to Ferdinand of Austria, GranDuke of Würzburg.