Yeah, heh, doubt "Cobblepot" is exactly Sicilian.
Don't know that you have to get into any British background exactly (actually really glad they didn't go with that cockney-accent interpretation here, always hated that for some reason), but yeah, he should clearly be an outsider of sorts within the high-class mafia circles. Whether that's simply economic, like he grew up poor and unconnected but rose through the ranks with his effective ruthlessness his bosses could acknowledge/reward, or it's him not being "legit mafia" due to no Italian background, it works either way. Kinda thinking it's more the former, but a throwaway line from a rival underboss about him being "some Limey's kid" or something wouldn't go astray either.
Art, I'd kinda question the logic there. Just because he went way out-of-the-box with Riddler doesn't mean he'll approach everything & everyone that way. There's a lot of classic iconography in this already as it gets, even Movie 1. Whether it's the oldschool 60s/70s influence on the suit's silhouette, or the car, or Catwoman being pretty "classic". Gotham itself with all the art-deco stuff, maybe even blimps from what we're hearing.
If it makes sense for his vision I think he'd "upgrade" Penguin no problem, the hat & monocle & umbrella gun, or at least some of the three. Could be as simple as "Riddler's overall look is admittedly a little goofier than most - we need to retool it given the tone here". That doesn't apply to everyone necessarily. For example, I could totally see Two-Face in this universe having one half of his business suit "conservative lawyer court attire" and the other side all "cliche pinstriped 40s mob boss".
That works with Two-Face because he's off his rocker. The monocle & hat applies to Penguin because, yes, character-wise he's a vain insecure guy trying to artificially climb the social-status ladder, who'd never be taken seriously if it weren't for the fact he'd snuff anyone who called him out over it.
The look we're getting works just fine here, it's perfect for "Oswald" proto-Penguin. But we've seen Reeves hasn't shyed away from using the name "Riddler" or at most of the iconography other than the costume (which again, I'd argue is more just because bright-green-leotard-or-suit-with-domino-mask-and-bowler is at the sillier end of the Rogues Gallery looks). Can't see why he'd have any problem having Oswald embrace the "Penguin" schtick, like maybe it starts out as an insult/mockery from people that he decides to own and run with. Sorta like TDK Harvey.
Just, like, keep it to scenes where he's out on the town mingling with the rich & famous, playing himself up as "legitimate". Doesn't mean he can't look the way he does now in the first movie while just hanging out at his club day-to-day, or in gunfights or whatever. But this is a guy who runs in fancy circles, at least once he's kingpin - you can have a scene or two with him looking all dandy-dude. There's a sensible middle-ground for this.
*Holy rambling posts Batman. Got a little carried away there. Hail Satan.