The API that Zotero provides which Overleaf uses does not have access to the database in which BBT keeps its keys1. There are two ways to get your keys into Overleaf:
- In Zotero, pin your BBT keys (right-click the items in Zotero, option
Better BibTeX ->
Pin BibTeX key. This will write them in the
extra field of your items, and those keys will no longer change when the item does. The zotero API will use the keys it finds there. This will bring your keys to Overleaf, but not the BBT-generated bib(la)tex.
- Auto-export the file using BBT to a folder that is synced to Overleaf using Dropbox, Google Drive, or their git support. This will bring the full BBT-generated bib(la)tex file to overleaf, which will include your citekeys.
You can also have BBT generate keys that are usually identical to the keys that the Zotero API gives to overleaf by default, so that the keys you have locally and the keys you see in Overleaf will usually be the samen. But the process by which the Zotero API generates these keys may vary the keys it offers per item sometimes, so this cannot be guaranteed.
1 by necessity. BBT is not allowed to add extra fields to the Zotero database, and they wouldn't sync anyway. A change in Zotero is planned that will add a formally supported citation key field to Zotero items; BBT will put its keys there instead of its own database and workarounds such as those listed here will no longer be required.