I don't think that's quite the right assessment. The membership didn't decide to 'lurch left' so much as it lashed out and cast a protest vote against the vacuity and emptiness of Burnham and Cooper. Cooper's bid for the leadership was entirely policy free until the last two weeks of the process, for example. The Labour membership is (or was, pre-expansion) quite willing to countenance candidates to the right of the party - but only if they have the competence and policy base to be convincing. I think the overwhelming perception of the Labour Party on seeing Burnham and Cooper was: 'with Blair, we went to the centre, but we could win. With these chumps? We'd move to the centre and still lose. What's the point?' and just picked Corbyn as a kind of fuck you to the bland, anodyne candidates that came forward.
Cooper still has... exactly that same problem. She's popular within the PLP by all accounts, but amongst the membership she's still seen as bland and lacklustre. There's a small clique of about 24% of the membership by the last internal Labour poll that rate her highly, and that's not enough to be winning anything.
I don't think she has any serious chance. The next Labour leader will be, at the very most, from somewhere on the soft left. The most 'right' you're going to go is probably someone like Keir Starmer, who isn't even that 'right' by Labour standards (definitely on the soft-left of the party), but has won grudging respect from the Corbynistas by being willing to slum it out in the cabinet. Loyalty goes a long way when convincing people to change their minds - respecting their decision even if you disagree with it goes a lot further than open hostility.
The real race is going to be decided by who can consolidate that soft-left support without alienating the Corbynites. The candidates in good position to do that are Keir Starmer, Lisa Nandy, maaaybe Clive Lewis (if he doesn't lose his seat, and if he manages to be slightly more conciliatory), and maybe Angela Rayner as a dark horse candidate.
I think Lisa would win if she goes for it. I mean, I wouldn't give even odds or anything, but she's my favourite.