Aha! Something interesting looking at a Richard Prince in horse-racing:
http://www.rootschat.com/links/01pbj/This book, about Paris and environs, published in 1845, describes St. Germain-en-Laye, just west of Paris. Mention is made of a racing-stud established in the area, with which "Mr Prince of Newmarket" was associated.
Could this be how Elizabeth and Alphonse met? It would be worth expanding the search from Paris proper to some of these areas near Paris.
https://archives.yvelines.fr/article.php?larub=24&titre=registres-paroissiaux-et-d-etat-civil (general link)
https://archives.yvelines.fr/ark:/36937/s0053fc3bd997cfb/53fc3bd9bec19 (hopefully direct link)
- a son Henry was born in 1843, at the residence of his parents in Saint-Germain, rue de Tourville, son of Richard Prince (English), aged 43 and his wife Phebe Hobbs, aged 42.
Unfortunately no sign of Elizabeth's marriage(s) or Louisa's birth in that area.
From
https://www.britishnewspaperarchive.co.uk/ in around 1851 Richard Prince was hired by the Marquess of Waterford so there might also be an Irish link...