The fact that Zero is a Britannian will set off alarm bells, because why would a teenager attack his own country? Ligth will know there is something up and will try to do further research. Like, he can send Lelouch's photo to Mikami or Misa to use their shinigami eyes or contact the Britannian Empire to spill out any info they had on Zero. The last one is important since Villeta, Cornelia, and Suzaku know who Zero is. Lelouch will just kill off Villeta (since Diethard and Sayoko already know about their relationship with Ohgi) and Cornelia (since he has her hostage). The issue is Suzaku, but he is likely not to spill the info since R2 Suzaku, despite hating Lelouch still views him as a friend, and assuming this R2 Lelouch then Suzaku still thinks he got his memory wiped and is thus not Zero. The OSI only answers to Emperor Charles, so unless Ligth keeps him alive (unlikely since he is likely to kill the Royal Family in a Death Note X Code Geass world), then they won't share info as well.
Ligth does have one lead, tho: Zero's actions. Knowing Lelouch, he will 100% target the NPA chief to know the identities of the Kira Task Force members, aka his only lead against Kira. Mello did this in canon against Ligth, who will 100% warn the NPA to secure the chief, or even threaten Britannia to help him. While Lelouch should bypass this via Rolo, Sayoko, his Geass, and tech, reports of police forgetting their memories or being unconscious will set off alarm bells. Ligth will also be investigating Zero's past exploits like the Orange Incident or the 1st Black rebellion, which have sus stuff. I'm taking about the existence of "Orange" and Jeremiah reacting to it in live TV. By controlling Britannia (which he can do via threats like he did to the US in canon), he will know Orange is a lie and that Jeremiah's actions are unnatural. During the Black Rebellion, a bunch of section blocks were disabled, something only Britannians can enter. Ligth will begin to suspect Zero has some sort of power like him because of this.
After that, Ligth just protects the task force members via the police. Too bad Black Knigths can easily beat them. Ligth can try controlling the Black Knigths tho....like Kallen who knows who Lelouch is. Ligth can also threaten the Black knigths to force Zero to reveal his identity. So Ligth has 2 wincons against Lelouch.
Sorry for the yap, I just liked discussing stuff like this.