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.
Amabel discusses (see Thm 2.20) Koszul duality of operads and cooperads in spectra (Ching has shown 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 and Ching's thesis. According to Ching and Harper, this bar adjunction is also an equivalence if you assume only that the operad is connective -- which again seems like a much weaker assumption than I expected!
Lots more good stuff in here...