Short Answer: Fight the rest of his villains/ Have an identity crisis
The first Superman movie, he fights Lex Luthor. In the second one, he fights Zod. By the third one, he is already fighting his evil self. And then in the fourth movie... Lex again.
In Superman Returns? Lex again, again! Man of Steel? Zod again. And then in Dawn of Justice... he fights Batman-- just kidding, it was Lex behind everything the whole time. Surprise!
But here's the thing. In his TV shows and comic books, Superman fights... and you're not going to believe this... OTHER bad guys. I know! Mindblowing.
So, um, can he fight them in a movie?
Please?
There are Brainiac, Bizarro, Parasite, Mr. Myxzptlk... that's four more movies right there.
Batman gets to fight all his bad guys. Spiderman gets to fight all his bad guys... at once! So why does Superman only get to fight either someone with no powers, or someone with his same powers?
And, while we're at it, let's not go back to the well on kryptonite again as his weakness. I know, crazy! But Supes is also vulnerable to magic. So... work with that. Why does Marvel get to own magic as an idea? DC has Dr. Fate, Zatanna... and magical baddies, too.
The more interesting answer comes from a conversation I had with a friend. One of us said that, really, he IS Superman and Clark is the alter ego. And other said nah, it's like it always says-- he was raised Clark and IS Clark, so Superman is the alter ego.
But then we realized the TRUTH: that he IS Kal-El... and BOTH Clark and Superman are personas he adopts to exist in different situations. He NEVER gets to just be himself, except when he is alone. When he is with anyone else, he had to pretend-- either to be a normal earthling, or a Justice Leaguer.
So THERE's your movie-- the decades of never being allowed to be himself have worn on Kal-El, and he's having a mental breakdown. Think about how that would resonate with today's audiences!
And wouldn't fighting one of those afore-mentioned villains bring him to that state? They have mental-bending, mind-screwing, power-sucking, perspective-blasting powers... any of them could make him-- force him-- to confront who he really is, and how he's always been a fraud to everyone he's ever known. Even Lois.
(And... maybe we found the Superman movie Nic Cage was meant to be in all along? Who portrays mental breakdowns better?)