This is after saving the HTML file and editing it to change the “1” to a “2” on Line 481? And checking that the KDF Iterations matches the kdfIterations value from the .log file?
The documentation is unclear, but I have never been able to find a data.json file for the browser extension. I know that some (or all?) of the vault data is stored in the .log files, but I don’t know if this is literally a log, or whether this is also the data that the browser extension reads back in when unlocking the vault.
Probably what I would try if I were you would be to restore all the contents of the extension data folder (e.g., on Chrome: %LocalAppData%\Google\Chrome\User Data\Default\Local Extension Settings\nngceckbapebfimnlniiiahkandclblb), then disconnect the internet and see if you are able to unlock your browser extension.
However, I thought you were already able to export from the browser extension on your “main PC”. What additional information are you hoping to recover from the Synology backups? Even if you are able to retrieve a backup of your local vault cache, there is no mechanism for pushing that vault back into the cloud — you can only use it for purposes of exporting or otherwise retrieving vault data (to then reconstitute into a new vault).