Path integral and Out-of-time-ordered (OTOC) correlator
The correspondence between the operator formalism and the path integral formalism is, in general, not fully understood and the issue of ordering prescriptions remains something that one may understand in specific cases but for which no general method is known. The "equivalence" of the path integral and operator formalisms is of a formal nature and has to discard/arbitrarily deal with several terms related to ordering ambiguities along the way, see also Qmechanic's answers here and here for more details and examples.
The path integral in the Keldysh-Schwinger formalism is a standard tool in non-equilibrium QFT, cf. e.g. Berges' "Introduction to non-equilibrium QFT".
I think this is what you are looking for:Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace
As mentioned the Keldysh formalism is the way to go. In this work they aim to provide a more mathematically elegant formulation of such formalism by means of a set of underlying BRST symmetries. It also has a chapter on computation of Out-of-time-order correlators.