Get a list of collections using:
https://api.zotero.org/users/[USER ID]/collections?key=[API KEY]
Where:
[USER ID]: Listed here: https://www.zotero.org/settings/keys[API KEY]: Create one here: https://www.zotero.org/settings/keys
Find the "key" attribute corresponding to the collection you want to use. This is your [COLLECTION ID].
Then, in Overleaf, create a new file. Select "From External URL", then insert the following URL:
https://api.zotero.org/users/[USER ID]/collections/[COLLECTION ID]/items?format=bibtex&key=[API KEY]
Name it whatever you want (with a .bib extension of course). Then create it. Whenever you add new items to the collection, sync the collection in Zotero, select the file in Overleaf and hit Refresh. Easy.
This should not be a paid feature.