I couldn't find a word of god on the matter, so there is no definitive answer to the question. However, here is what we can infer from Suicide Squad and Batman v Superman:
Lex Luthor is not a good fit for a Suicide Squad
- First of all, the Suicide Squad, as shown in the movie, is a front line "military" unit. Its members are an expert marksman, a boomerang thrower, a giant lizard, a flamethrower, etc. Lex Luthor is not a fighter and his talent lies in his ability to manipulate people in the shadows.
- The members of the Suicide Squads are expandable criminals that no one cares about. If they die during a mission, Waller could easily cover it and the general audience would never know or care. On the other end, Lex Luthor is a famous billionaire that made the headlines a short time ago...
- Lex Luthor is too dangerous: the plot of BvS is that he manipulated the whole world and succeeded in having Batman, and the US government, believe that Superman is a threat that should be eliminated. Who can know what he could do if joining the squad?
- Amanda Waller likes nothing more than controlling her pawns. For street-level criminals like the members of the Suicide Squad, a bomb is sufficient. But Lex Luthor is a genius and has almost unlimited wealth. If the Joker can find a way to inactivate the bomb, Luthor is likely to do it too.
- As you said, Luthor is crazy like hell and believes the apocalypse is upon us. Sure, the sanity of the other Suicide Squad members is questionable, but even Harley Queen can be controlled in some fashion. How do you coerce a demented Lex Luthor to go after dangerous meta-humans?
- The main reason that explains why Superman was not a threat, after all, is that he "shares our values", that is he is somehow human. The members of Suicide Squads are not good guys but if a planetary threat arose, they would probably fight it, as they have a family, etc. We know from BvS that Lex Luthor is a diehard, and does not hesitate to use everything he needs to reach his goal. The Suicide Squad was created in case the "next Superman" is a danger for mankind, Lex Luthor created Doomsday from General Zod, who is literally the "next Superman".
She may being secretly using Lex Luthor for other purposes
That's pure speculation, but we know that Amanda Waller has an extended knowledge of meta-humans. The only other person we know to be so knowledgeable on the topic at the beginning of Suicide Squad is Lex Luthor. Maybe Waller got her information from him. In addition, we don't know what Waller knows about Darkseid's incoming threat. If she knows about it, then Lex Luthor is more valuable to her as a source of information rather than fighting "zombies" in Midway City...