Let's see.
First, there aren't federal libel laws. There is exactly 0% chance that Congress is going to add one just because Donald Trump's feelings get hurt when the media tells the truth about him, much less one that would allow him to successfully sue them. Even if Republicans voted 100% lockstep in favor, they would not have the votes. They don't have 60 votes in the Senate and no way even a single Democrat would vote for it. Not that all Republicans would be in favor anyway. Plenty of them are smart enough to know that such a bill would bite their side in the ass as it would also allow Democrats to go after right wing media. Trump can't even get Congress to fund his stupid wall, but he is going to get them to do this? Come one.
As for the Supreme Court, they have made rulings that make it basically impossible for politicians to successfully sue for libel. While it is theoretically possible for them to overturn this precedent, they would first need a case to do it. Like Congress passing a law and it getting challenged. Which as I already explained, is not going to happen. And even if they did, what reason do they have to change it, because an unpopular president wants them to? Please. These people have lifetime appointments, Trump has no leverage over them.
Seriously, this is not going to happen. I suspect that Trump told his people to look into it. And they did and found there was no way to do it or already knew. But Reince can't well go on TV and say that his boss is a petulant idiot who wants to implement an unconstitutional plan that won't happen.