Lex wasn't in Batman: TAS. He was one of the two overarching Big Bads of Superman: TAS (the other being Darkseid) and was a Corrupt Corporate Executive
, albeit one who was also a scientific genius. He got exposed early in Justice League and spent its first two seasons a straight-up supervillain/ Mad Scientist
before getting a pardon for helping beat the Justice Lords. For the first season (or two seasons) of Unlimited he became a politician, ran for president, and was ultimately the mastermind behind Project Cadmus (though Brainiac
turned out to be the mastermind behind him
). Finally, in the last season of Unlimited he went back to supervillainy after being exposed again, and ended up running the Legion of Doom
And there's a brief history of DCAU-Lex, who generally did a good job at combining the disparate aspects of the character into a whole that made sense.