There's actually some debate about this in a GA-based campaigning group after someone posted
this. My response was really long-winded, of course, but for here I'll just say:
Unless LA maintained the 50% run-off element (which only GA & LA to my knowledge have) when transferring to party-only primaries, then Vitter would have definitely won in the primary election. If there was a primary run-off like there is in GA between the top two in the party, then it's certainly possible that Dardenne or (more likely) Angelle would have won. However, that's not guaranteed and I think Vitter's advantages would have overcome his disadvantages in the end.
About all that would have ended Vitter's chances would be ranked-choice ballots and instant-runoff voting, which was one of the premises in the article that said LA elected the "wrong" Governor. However, that ain't happening, so...I think Vitter would have likely won regardless in any realistic scenario involving traditional US voting.