I have finally found a source which puts together the pieces in a satisfactory way, at least in the stable setting, here:

Amabel, Araminta. "Poincaré/Koszul Duality for General Operads." arXiv preprint arXiv:1910.09076 (2019). [latest arxiv version](https://arxiv.org/abs/1910.09076).

 - Amabel discusses (see Thm 2.20) Koszul duality of operads and cooperads in spectra (Ching has [shown](https://arxiv.org/abs/1009.5034) that the bar/cobar adjunction for spectral operads / cooperads is an equivalence here with no conditions!!! This is very surprising to me given that most authors assume very very restrictive conditions like being a quadratic operad to get similar results.)

 - Amabel also discusses (see Thm 2.21) the relationship between Koszul duality for operads and cooperads, and the induced bar/cobar adjunctions for the (ind,nilpotent,with divided powers co)/algebras of the corresponding co/operads, with reference to [Francis-Gaitsgory](https://arxiv.org/abs/1103.5803) and [Ching's thesis](https://arxiv.org/abs/math/0501429).

 - Lots more good stuff in here...