This must be theatre's version of Godwin's Law. Bertolt Brecht's mobster Arturo Ui is, famously, a sly spoof of Hitler – a backstreet crook who wrings power via protection racket; the Third Reich shrunk down to the small-time. By looping in one Donald Trump, Bruce Norris' springy new adaptation seems rather too pat – at once overblown and underpowered.

Ui's rise mirrors Hitler's step-by-step – from the extortion of a respected local leader (Michael Pennington's Dogsborough resembles the ghost of Bernie Sanders) to the annexation of a nearby cauliflower market, via a range of arson, intimidation and indoctrination. He starts clumsy and uncouth, little more than a dumb bungler, but, coached by an old-school actor, sculpts himself into a strongman leader: his walk, a goosestep in miniature; his stance, a salute that folds itself into familiar crossed arms. Brecht underlines the villainy with shades of Richard III, as Ui woos a widow over her husband's coffin and faces down the ghosts of his victims.

Norris' contemporary version is soaked in a thousand mob movies, almost a spoof of a spoof of a spoof. Ui's hoodlums, therefore, are lickspit stereotypes: Giles Terera's Ernesto Roma, the dead-eyed, hot-headed psycho, jittery with bloodlust, Joe Pesci-stylee; Lucy Ellinson the chuckling high-pitched joker and Guy Rhys, the calm, cold-blooded killer. The whole thing is played with a rubbery, cartoon violence – big cans marked kerosene and guns spun around fingers. Simon Evans instils the feel of a Simpsons episode: every citizen a caricature with a comedy voice. Lawyers gurgle and quiver, judges boom. Reporters speed through their scoops.

In the middle of all that, Lenny Henry's Ui becomes almost a straight man. Beneath a back-and-sides so short, it's non-existent, he's a huge presence – lumbering at first, then genuinely imposing. His mannerisms, like Norris' script, glance off The Donald – no all-out Alec Baldwin impersonations, but the odd echo here and there. He'll address us as "folks," or let slip a crude boast: "No-one knows the system better than me." By the end, he's addressing a crowd ("Biggest ever. Fact.") beneath a banner proclaiming: Make This Country Great Again.

Subtle, this ain't. It was never going to be. In fact, it's almost exactly what you'd expect and that obviousness makes it seem rather toothless. Where lampooning Hitler in 1941 was a weapon in itself, Trump's rather impervious to the approach. He's already a joke, and, leaving aside the horse having bolted, the show struggles to reconcile the laughing stock loon with the very real danger. It's all rather safe: a well-worn joke we know and love. Trump's too easy a target. In Turkey, Recep Tayyip Erdogan's following the Ui playbook scene-by-scene, but Trump's by far the better butt. Laughter almost lets him get away with it.

And this is, first and foremost, a fun night out. Turning the Donmar into a rusty, run-down speakeasy, Peter McKintosh's design looks a few billion dollars, and it's playfully lit by Howard Harrison – all sodium glare and streetlamp shadows. Between scenes, we get lush snippets of semi-relevant pop songs: Radiohead's "Burn the Witch" and Rag'n'Bone Man's "Human".

Evans folds the audience into the mix, pulling volunteers onstage as objectors then bumping them off. Ticklish as it is to see total innocents bemused by charges against them and bandaged beyond all repair, it does skew the show further towards sketch comedy. A stilted play at the best of times, Arturo Ui needs all the momentum it can muster. His unstoppable rise is a runaway train. Our involvement – and our embarrassment – makes it stall.

Evans' point is that politics is all theatre – and we willingly play along. It's telling that Ui's boys aren't heavies, but clowns, too entertaining to scare us off. The interaction earns its keep right at the end, as Ui calls a vote on new executive powers. Either we stand in support or we step out onto the stage in protest – and so make a spectacle of ourselves. Staying seated counts as abstention, not resistance. All it takes for evil to win is for good men to do nothing – and for theatres, perhaps, to stage soft-centred satire.

The Resistible Rise of Arturo Ui runs at the Donmar Warehouse until 17 June.