There has been some work along these lines showing that provably recursive functions of various theories are exactly those in some hierarchy up to the prooftheoretic ordinal of the theory For example the paper Fairtlough Matt Wainer Stanley S Hierarchies of provably recursive functions Handbook of proof theory Stud Logic Found Math  NorthHolland Amsterdam  briefly discusses the case of Peano arithmetic with transfinite induction added but I would guess people have by now got as far as the constructive ordinals you mention My impression is that you should  in principle be able to get up to any given constructive ordinal but there is no honest way to get all the way to the ChurchKleene ordinal