The governor knows his claims are foolish and he knows he's building a controversy out of thin air. It's playing great with his constituents, and the fact that his position of power inclines people to take him seriously means he can get away with it.
If your premise is true (that the harder those who understand this push against it, the harder he will push back, and the voters of his state like him for that) then there is no winning strategy here.
Doubtful.
This is absolutely one of those cases where public attention and pressure can spare someone from getting lost in the legal system. Burying the story only helps those abusing their power.
His PAC is running ads like this. https://www.youtube.com/watch?v=9IBPeRa7U8E
To a lot of us, that looks like nonsense, but you have to remember how far through the looking-glass a lot of the conservative populace has gone... The press is now the enemy in their mind, and they don't understand technology well enough to get why these claims are ridiculous. So he gets to build steam and any pushback is just seen as more "fake news" coverage that would (of course) support their fellows in the media.
It's deeply frustrating because it's a form of social jujitsu built atop existing mistrust, and there aren't good ways to combat it.