The dgruyter.pdf companion to the dgruyter.sty available from https://www.degruyter.com/page/production-for-authors (specifically this .zip archive, if you want to obtain the .sty file from the files in the archive, run pdflatex dgruyter.ins) has the following to say about bibliographies
It is recommended to use the standard bibliography mechanism. You might copy
and paste your bibliography entries from elsewhere into the thebibliography environment or, more elegantly, use BibTeX. The dgruyter package does not prescribe
any particular bibliography style.
The natbib package can be loaded additionally, so an author-year-style bibliography layout is possible. The special citing commands \citet, \citep and so on can
be used. Feel free to configure natbib, e.g. with \setcitestyle{numbers} in your
document preamble to force the numerical mode.
Alternatively, the biblatex package can be used. Be aware, however, that there is
an abundance of options which have not all been tested for compatibility with the
dgruyter package. The following seem to work fine for the numeric style:
\usepackage[backend=bibtex]{biblatex} to load the package,\addbibresource{BIBFILE} to load the .bib-file,\printbibliography[env=bibnumeric] to output the bibliography.
With author-year citation you have to skip the optional argument of
\printbibliography.
So I would try to avoid biblatex if possible or at least try to clear the usage of biblatex with the responsible editor (or another contact at de Gruyter).
There is a confusion about BibTeX styles and biblatex styles here, so we must shortly address some terminology.
The big two ways to produce bibliographies semi-automatically in LaTeX are with BibTeX and biblatex. You can read much more about their pros and cons at bibtex vs. biber and biblatex vs. natbib and a bit about the history (as I understand it) at A design question: citation commands.
The important bit for us is that bibliography styles for BibTeX-based solutions are given in a .bst file and loaded with \bibliographystyle (the citation style is controlled by \usepackages like cite, natbib etc. in combination with a compatible .bst file). Such that a document using the BibTeX style plainnat with natbib looks roughly like this
\documentclass{article}
\usepackage{natbib}
\begin{document}
A bare citation command: \citep{sigfridsson}.
A citation command for use in the flow of text: As \citet{sigfridsson} said \dots
\bibliographystyle{plainnat}
\bibliography{biblatex-examples}
\end{document}
For biblatex the bibliography style is implemented in a .bbx file and the citation style in a .cbx file. These files are loaded with the style option (or separately with bibstyle and citestyle) when you load biblatex. An example using biblatex's numeric style would look like
\documentclass{article}
\usepackage[backend=biber, style=numeric]{biblatex}
\addbibresource{biblatex-examples.bib}
\begin{document}
A bare citation command: \citep{sigfridsson}.
A citation command for use in the flow of text: As \citet{sigfridsson} said \dots
\printbibliography
\end{document}
You can read more about the transition from BibTeX to biblatex at What to do to switch to biblatex?.
The file bzrp.bst you link has the file ending .bst and one would therefore be inclined to think it is a BibTeX style that is used like plainnat in the example above. But if one opens the file and examines it, the first line one finds is
\ProvidesFile{bzrp.bbx}[2008/01/04 bzrp biblatex bibliography style]
and indeed, the remaining contents of the file make it clear that this is actually a biblatex style. So the file must be renamed to bzrp.bbx and then used like numeric in the biblatex example above. Since the style is lacking a companion .cbx citation style it probable needs to be used at least like
\usepackage[bibstyle=bzrp,citestyle=authoryear]{biblatex}
this comes close to what your colleague suggested (minus some other settings).
A first attempt from your code above (I removed packages that were not relevant for the example, moved babel to before dgruyter to avoid an option clash error and remove the global ngerman option and replaced the non-standard \sqcitet by \textcite) could thus be
\documentclass[openany]{book}
\usepackage[main=french,italian, spanish, portuguese, latin, greek]{babel}
\usepackage[small]{dgruyter}
\usepackage{microtype}
\usepackage[autostyle=true]{csquotes}
\usepackage[bibstyle=bzrp,citestyle=authoryear]{biblatex}
\addbibresource{biblatex-examples.bib}
\begin{document}
Or, déjà au 18\textsuperscript{e} siècle, Pierre Bonamy a affirmé
que l'origine des parlers romans
\blockquote{n'est autre chose que la langue [latine]
parlée et employée dans les discours familiers}
(\textcite{cicero}, 586, cité par \textcite{sigfridsson}, 596).
Cette approche représentait un grand pas en avant pour la romanistique,
mais ella a dû attendre le 19\textsuperscript{e} siècle, avec \textcite{worman},
82 comme précurseur, pour accepter, en théorie,
que les langues romanes remontent au latin parlé.
Peu à peu, les romanistes en sont arrivés à convenir qu'en
raison de la différence notable entre le latin écrit et le latin
parlé dans l'Empire romain, due à l'évolution plus rapide de ce dernier,
les langues romanes sont la source principale d'accès à ce qu'on est
convenu d'appeler le \textit{latin vulgaire} (cf. \textcite{geer}, 4).
Cependant, la pratique lexicographique a dû attendre l'année 2008
(*/\textquotesingle{}kad-e-/: 08/08/2008) pour voir apparaître
autre chose que des lexèmes en graphie du latin écrit de l'Antiquité
comme étymons des lexèmes héréditaires romans
(cf. ci-dessous et cf. \textcite{nussbaum}, 145, 153).
\printbibliography
\end{document}
Unfortunately, if you try to run this code on a relatively recent LaTeX system, you will get errors like
! Illegal parameter number in definition of \blx@defformat@d.
<to be read again>
3
l.15 #1\ifblank{#3
}{}{\addcomma\space #3}
That is because the file predates the changes in biblatex 3.3 that are discussed in Biblatex 3.3 name formatting. (Indeed, the file header suggests the file was last changed in 2008, more than twelve years ago and even well before the first stable release of biblatex 1.0 in November 2010.)
bzrp.bbx is completely outdated and not compatible with a current version of biblatex. It might be possible to change it as explained in Biblatex 3.3 name formatting so that it compiles again with current biblatex versions, but biblatex has progressed a lot since the style was written, so that might not be enough. It may be easier to just start anew. I could not find any mentions of this style online, so I can't help you with contacting the author for support, I'm afraid.
The following compiles without any issues
\documentclass[openany]{book}
\usepackage[main=french,italian, spanish, portuguese, latin, greek]{babel}
\usepackage[small]{dgruyter}
\usepackage{microtype}
\usepackage[autostyle=true]{csquotes}
\usepackage[style=authoryear]{biblatex}
\addbibresource{biblatex-examples.bib}
\begin{document}
Or, déjà au 18\textsuperscript{e} siècle, Pierre Bonamy a affirmé
que l'origine des parlers romans
\blockquote{n'est autre chose que la langue [latine]
parlée et employée dans les discours familiers}
\mkbibparens{\textcite[586]{cicero}, cité par \textcite[596]{sigfridsson}}.
Cette approche représentait un grand pas en avant pour la romanistique,
mais ella a dû attendre le 19\textsuperscript{e} siècle, avec \textcite[82]{worman}
comme précurseur, pour accepter, en théorie,
que les langues romanes remontent au latin parlé.
Peu à peu, les romanistes en sont arrivés à convenir qu'en
raison de la différence notable entre le latin écrit et le latin
parlé dans l'Empire romain, due à l'évolution plus rapide de ce dernier,
les langues romanes sont la source principale d'accès à ce qu'on est
convenu d'appeler le \textit{latin vulgaire} \parencite[cf.][4]{geer}.
Cependant, la pratique lexicographique a dû attendre l'année 2008
(*/\textquotesingle{}kad-e-/: 08/08/2008) pour voir apparaître
autre chose que des lexèmes en graphie du latin écrit de l'Antiquité
comme étymons des lexèmes héréditaires romans
\mkbibparens{cf. ci-dessous et \textcite[cf.][145, 153]{nussbaum}}.
\printbibliography
\end{document}
but of course it doesn't look quite like what bzrp.bbx would give you.
