Georgine offers nothing that Georgina doesn't do better. With a French accent, it sounds OK but would people say it that way? I doubt it.I like Georgina very much, with Gene or Genie as nns if necessary. But if I wanted something sounding like Genie, I'd just use Jean.
I think it's really sweet! I prefer Georgina, though I'd be delighted to encounter a Georgine IRL.I'd use Eugenia, one of my favorite names, as a full name for Genie.