Apparently the Prime Minister believes that parliament “cannot operate like some gentleman’s club where the members make up the rules and operate them among themselves”. But if so then that's only because there ain't sufficient gentlemen (and ladies) at Westminster. Parliament should be capable of operating like any decent, self-respecting gentlemen's club. That it isn't is why we're in this situation in the first place.
And now, in the crazy rush to react to public anger, the Commons seems certain to push thrugh reforms that make little sense and, most probably, won't address some of the major issues anyway. Certainly, the idea of some external "regulator" is a terrible one. The Prime Minister has surely heard of "regulatory capture"? Why would this not, eventually, happen with parliament too.
In any case, we already have a kind of regulator: it's called the electorate. There shouldn't be any need for any other body.