I suspect that what's really 'got' the Corbynite faction is not the minimum wage or any other policy, but the constitutional change.
If you look at the history of the last 50 years the constitution of the Labour party is by far and away the most important issue for the Hard Left. It isn't workers' control or public ownership, it isn't taxation or pensions, it isn't full employment, education or even the NHS. What has really galvanised men like Corbyn, McDonnell, Lansman and even Tony Benn in his day, are questions like 'Who gets to elect the NEC?', 'How are local MPs selected and how can they be deselected?', 'How is the leader of the Labour party elected'?', 'Who gets a license to have a stall at the Labour Party Conference'. This is the stuff that gets them out of bed in the morning. The other stuff about the NHS etc pales into insignificance compared with these epochal questions.
The exception to this constitutional navel-gazing is of course international policy. Palestine, NATO, Cuba, Venezuela. Limitless amounts of time are traditionally spent discussing these issues - a sign not of any commitment to Labour's traditional internationalism but rather of the Far Left's morbid feeling that socialism in the West will never happen and therefore it's nicer to think of faraway places where it might. Defeatism in other words.
With Starmer on the offensive now over the very issue that means most to them they have decided to go for broke. I'm starting to think that Starmer understood this and is simply pushing their buttons, hoping to spark a battle royale. Certainly nothing would make him more popular with the country at large than a Kinnock-like slaying of the Hard Left at party conference.