As for the "will/should/could Batman show up" question, I'll just say this:
Are you kidding?
1. Of course he will and our dear DC Studios CEO basically inadvertedly confirmed as much last year:
2. Not only should he show up, but I'll go as far as to say that the way that this show uses Batman will be what reveals whether this show justifies its existence or not. And by that I mean: If they were to use him in a Luke Skywalker in Mandalorian fashion where he shows up only at the end for a "OMG BATMAN!!!!!!!!!!!!!!!!!" moment that'd be so, so ****ing lame I don't even have enough words to describe it.
The fact of the matter is that in the world that Matt created, he showed us a Batman that deals primarily with threats of the criminal underground and that not only that, ended the movie with this lovely conversation that directly foreshadowed the events of this show:
So he ended the movie setting up the fact that there'd be a power grab gang war in Gotham and that not only that, Batman would have to deal with it (which... duh!!!! What else is he gonna do? Organized crime is what lead to that mess in the first place. If anything it'd have be his #1 priority at this point) and what is this show about? That exact power grab. That exact gang war in which every single Gotham City gang and mob boss is gonna be participating.
I'll make this very clear:
There is no way to justify Batman being absent for all of the show except the finale. None. Within the universe Matt created it's impossible to justify this show happening and Batman not showing up in at least a couple episodes.
And unlike, Titans, Batwoman, Gotham or any of the "non-Batman" shows that came before, there's not even a good excuse for him not appearing. The show takes place directly in Gotham, in the criminal underground where Batman constantly fights. There is nothing you can say or do to justify him not appearing.
And you can't bring me the whole "But Pattinson's too busy" because he wasn't. The show began filming 2 months after Mickey17 ended and all throughout he was completely and totally free. Not only that but he was spotted in New York several times when the show was filming. Maybe you can say he'd be expensive but streaming hires expensive stars, guest stars all the time. So I'm not buying it.
Also I ****ing hate this idea that just because you use Batman in the show you're somehow taking away from Penguin. Batman is a crucial part of Penguin's story. He always has been. You cannot do a show about him where you're dissecting his character and not explore his relationship with Batman and how he evades getting incarcerated by Batman. Batman, in this show, would serve the same purpose every other character except Penguin serves: A side character. So no, just because you put him in this doesn't mean it'll suddenly become a Batman show. (Also I know people are gonna bring up Joker and Venom but Joker takes place BEFORE Batman was a thing, and Venom takes place in San Francisco and they completely removed Spider-Man from the equation in that take. Same can't be said for The Penguin, in which Oz already met Batman several times, takes place right in the middle of Gotham and right after the movie so there's no reason Batman would not keep his eyes on him)
What's the best way to use Batman here, then?
Well, maybe there are other ideas out there, but the way I'd do it, and the way I think they'll do it, is this:
Batman makes a deal with Penguin. Penguin agrees to serve him information or something about the criminal underworld in exchange for protection or something of that kind. Batman shows up every couple episodes in small scenes so Penguin informs him, and ocassionally steps in when things get particularly messy (I'd imagine things are gonna get /extremely/ messy in the final episodes, where a whole building gets blown up. I imagine Batman will be a lot more involved there and maybe we'll even see some action) Throoughout the show we mostly hear and are shown the consequences of him intervining on other criminal operations as a result of the information Penguin feeds him.
This would tie in nicely with the whole "Rata Alada" of it all. Oz does become a stool pigeon of it all.
But Oz abuses this deal.
He eventually uses it in his favor so he ends up becoming the new kingpin of Gotham. By the end, he is kingpin, and Batman has to deal with the fact of knowing he got outsmarted and played by Penguin, who he previously understimated. It's the true victory for Oz: He won over all his criminal adversaries, the police and of course Batman. Ties in his character entirely.
Hell maybe we've already seen what Batman's first scene in the show is...
And after he leaves, some poor sob overheard...
But that's just a theory...