Some interesting issues here: the official who wants to make a law restricting flags to government entities, would also allow Mardi Gras flags. Mardi Gras is of course a Roman Catholic religious festival that marks the beginning of Lent. Do other religious groups get to fly flags for their own events, or just the Catholics?
June is recognized as LGBT month by the Federal government. Maybe not by Louisiana, but then the Federal government outranks State. Isn't that a government entity? Citizens are not allowed to celebrate a government proclamation?
Does the Korean War veteran who initiated the complaint against the LGBT flag also complain about the flying of Teabagger flags, like their "Don't Tread On Me" snake? Can the Boy Scouts fly their flags at gatherings on government property, or should that be prohibited, too?
I presume a Confederate flag would be right out, correct?
What about corporate business banners & flags? Suppose the Louisiana employees of ExxonMobil rented a public park for a company picnic outing one weekend. Could they not raise a corporate flag, marking their site? I've seen that done before.
But above all, a flag is a form of free speech. And unless a US or State flag is struck from its designated flagpole without authorization, and replaced by a private flag, I don't see how flying the rainbow flag can be made illegal. Except, naturally, in Louisiana.