Hi there
You can designate that the sync should sync only a specific folder in the remote assets library rather than the entire assets library by using an API key (see How to share access to your Assets Library) sharing that one folder rather than the entire library.
Since the files are uploaded to your local Foundry data folder through the local Foundry server, though, it’s possible that some file types are not synced, e.g. zip files. Specific valid asset files that fail to sync will generally not hold up the rest of the files from syncing, unless something is going wrong. If that is the case, could I please ask you to open the dev tools console (F12) (before encountering the error) and then share a screenshot of any errors that appear there?