Thanks for the information -- it is important to know that your SQL Backup version is at 5.3 because the file browser behavior changed again in 5.4.
In 5.3 server components, the SQL Backup file browser uses a combination of a delegated credential of the SQL Server service account and the SQL Backup Agent service startup account, so both accounts need access to your network resource in order to browse it. In addition, the SQL Server account impersonation requires you to be a local administrator on the SQL Server if you connect the SQL Backup Console using Windows Authentication. If you connect using a SQL account, you need to set the "AllowSqlBrowsing" registry key.
In 5.4, the file browser impersonates the Windows user running the SQL Backup console, which brings in its' own different issues.