I'm pretty sure the prosecution can only appeal on matters of law such as if bad evidence had been allowed. Any decision based on the facts cannot be appealed, just like it couldn't be appealed in a jury trial (i think.).
Either way, even if the DPP did find something to appeal on, I doubt it would change the final verdict given how flimsy the prosecution case was.
Presumably an error of law would be the only basis of an appeal - as you suggest, improper admission/exclusion of evidence being one example. If it's treated like discretionary decisions are here, then you're right, it wouldn't simply be a matter of factual interpretation.
It's still more than one would normally get from a jury acquittal, which could be based on a non/mis-application of the law, or influence from a particularly unbalanced judicial charge, for instance.