I think The Penguin is a perfect self-contained story, but I do think there could be potential for a Season 2 *after* The Batman Part II.

The Batman Part II - The Penguin and The Court of Owls wreck havoc across Gotham. At the end of the movie, Batman sends The Penguin to Arkham.

The Penguin Season 2 takes place entirely in Arkham. It would essentially be that scrapped Arkham show, but with Colin Farrell's Oz as the main POV. Once again, Oz would be at the bottom and he would need to rise to the top of an even more insane social hierarchy. You would get Oz interacting with The Riddler, The Joker, and other rogues. We would also get the cat & mouse chase with Sofia and Julian we loved so much in Season 1. A mental institution would also be a great way to further explore Oz's sociopathy and pathological lying. Imagine a one-on-one therapy session between Oz and Julian or...Oz and Hugo Strange perhaps. (And this is getting too fanservicey, but Imagine a group therapy session with Oz and all the Rogues led by Dr. Jonathan Crane or Dr. Harleen Quinzel.) At the end of the season, The Penguin would organize a massive jailbreak and free all the Arkham prisoners, including Sofia, onto the streets of Gotham.

The Batman Part III - pure unadulterated anarchy - a mix of Knightfall, No Man's Land, The Long Halloween, and Arkham City.

This would set up The Batman Epic Crime Saga to have a nice thematic narrative of bringing genuine law-abiding order to Gotham.

The Batman Part I - dismantling The Falcone Family and organized crime
The Penguin Season 1 - transitioning from organized crime to old money
The Batman Part II - dismantling The Court of Owls and old money and privilege
The Penguin Season 2 - transitioning from old money to pure anarchy
The Batman Part III - dismantling pure anarchy and bringing genuine law-abiding order to Gotham