I see no reason to have Kryptonite in the next film. No Superman stories have ever been made better by the inclusion of Kryptonite. No Superman stories have ever been made worse by the exclusion of Kryptonite. Kryptonite is by far the least necessary aspect of the Superman mythos. When Denny O'Neil, one of the best writers DC has ever had, took over writing for Superman at the tail end of the silver age, the first thing he did was make Superman immune to Kryptonite. Kryptonite barely appeared in Superman comics throughout the late 80s and most of the 90s, to the point that there was a ten year stretch of the character's appearances where Metallo did not have a Kryptonite core, and instead relied on his strength and a later upgrade allowing for shapeshifting abilities to battle Superman. If you ask people to make a list of the top five best Superman stories ever, generally you're going to see Kingdom Come, All Star Superman, Whatever Happened to the man of Tomorrow, For The Man Who Has Everything, and What's So Funny About Truth Justice and The American Way? on such lists. Kryptonite is in exactly one of those stories, and even then it's barely in that one.
You don't need Kryptonite to write a Superman story with stakes and genuine conflict. All you need is talent.