Try the following MWE:
%!TEX TS-program = lualatexmk
\documentclass{scrartcl}
\usepackage{fontspec}
\usepackage[backend=biber,style=authortitle-comp,abbreviate=false]{biblatex}
\usepackage{biblatex-bookinarticle}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{plato:complete,
Author = {Plato},
Editor = {John M. Cooper},
Keywords = {Primary Source},
Location = {Indianapolis},
Publisher = {Hacket Publishing Company},
Title = {Complete Works},
Year = {1997}}
@bookinbook{plato:dialogue,
Author = {Plato},
Crossref = {plato:complete},
Pages = {156--191},
Title = {Dialogue},
Titleaddon = {\bibstring{bytranslator} Donald J. Zeyl}}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
These are the complete works: \cite{plato:complete}. They contain the following text: \cite{plato:dialogue}.
\printbibliography
\end{document}
It gives this result:
Is this result really intentional? I.e. do we really want the name "Plato" not to be printed again before "Complete Works" in the bibliography entry of the dialogue? Should it not better be like this:

