I have tried synchronizing my Zotero library with Overleaf's Zotero sync. Overleaf uses the Zotero Web Exporter API.
However, I have run into some issues with BibLaTeX that is supported by Better BibTeX but not by the Zotero Web Exporter API.
BBT supports field that are unsupported by Zotero, for instance acmid and articleno:

@inproceedings{Bird:2004:NNL:1219044.1219075,
location = {{Barcelona, Spain}},
title = {{{NLTK}}: {{The Natural Language Toolkit}}},
doi = {10.3115/1219044.1219075},
timestamp = {2018-10-31T10:24:07Z},
booktitle = {Proceedings of the {{ACL}} 2004 on {{Interactive Poster}} and {{Demonstration Sessions}}},
series = {ACLdemo '04},
publisher = {{Association for Computational Linguistics}},
author = {Bird, Steven and Loper, Edward},
date = {2004},
articleno = {31},
acmid = {1219075}
}

The Web Exporter incorrectly outputs these in a note:

@inproceedings{Bird:2004:NNL:1219044.1219075,
location = {Barcelona, Spain},
title = {{NLTK}: The Natural Language Toolkit},
url = {http://dx.doi.org/10.3115/1219044.1219075},
doi = {10.3115/1219044.1219075},
series = {{ACLdemo} '04},
booktitle = {Proceedings of the {ACL} 2004 on Interactive Poster and Demonstration Sessions},
publisher = {Association for Computational Linguistics},
author = {Bird, Steven and Loper, Edward},
date = {2004},
note = {tex.acmid= 1219075
tex.articleno= 31}
}

Have you considered including BBT in the Zotero Web Exporter API pipeline?