I have the impression that Lelouch's drive is caused by his perceptions about what is needed to justly order his world. Therefore, a key part of deciding his agenda in another world would depend on what he thinks of the status quo.
The just ordering of the world is a key political question. Thus, how the writer evaluates things can bring in RL politics, which may be more of a mess than you want to deal with.
One option, which might easily get too political, would be openly cooperating with one of the world powers, in exchange for help training and equipping his forces.
Another, less political, would be if he decides that because he plans to leave, he doesn't want to start any messes that he won't be around to manage later. Focus on the limited time, and whether it is worth causing destruction if one won't be around long enough to get it to pay off. (It's still potentially political, because there are folks who would be offended if he decides that a power, like say the United States, isn't as evil as Britannia and doesn't has as urgent a need for destruction. Even if he goes after, say, Mara Salvatrucha, it is still possible to tick people off.)
Lastly, training and education. There is a lot of things that can be learned in the one world that would transfer to the other. Then there is the luxury of time to study without being hunted, or having to be operational. So, professional development, languages, if those transfer. The ideal situation would be if the visitors learn, but don't age, and if time is stopped back home. There is never enough time for everything in military education and training, so 'bonus' time would be priceless, and whatever they have should not be wasted.
Then there are the opportunities that California provides that they might not have access to in their world. Yes, they probably don't have the documentation for passing background checks, but there is a lot that can be done without that. I dunno if they could pull off something with the California State Defense Force, but I imagine they would have a harder time with the California National Guard. Still, there is a lot of opportunity. Even and maybe especially without leaving the country to find terrorist training camps.
The just ordering of the world is a key political question. Thus, how the writer evaluates things can bring in RL politics, which may be more of a mess than you want to deal with.
One option, which might easily get too political, would be openly cooperating with one of the world powers, in exchange for help training and equipping his forces.
Another, less political, would be if he decides that because he plans to leave, he doesn't want to start any messes that he won't be around to manage later. Focus on the limited time, and whether it is worth causing destruction if one won't be around long enough to get it to pay off. (It's still potentially political, because there are folks who would be offended if he decides that a power, like say the United States, isn't as evil as Britannia and doesn't has as urgent a need for destruction. Even if he goes after, say, Mara Salvatrucha, it is still possible to tick people off.)
Lastly, training and education. There is a lot of things that can be learned in the one world that would transfer to the other. Then there is the luxury of time to study without being hunted, or having to be operational. So, professional development, languages, if those transfer. The ideal situation would be if the visitors learn, but don't age, and if time is stopped back home. There is never enough time for everything in military education and training, so 'bonus' time would be priceless, and whatever they have should not be wasted.
Then there are the opportunities that California provides that they might not have access to in their world. Yes, they probably don't have the documentation for passing background checks, but there is a lot that can be done without that. I dunno if they could pull off something with the California State Defense Force, but I imagine they would have a harder time with the California National Guard. Still, there is a lot of opportunity. Even and maybe especially without leaving the country to find terrorist training camps.