• The account which actually rights the backup is the engine, not the agent.

    You will need to make the SQL Engine service run as a domain account.

    You could try adding the computer account in on the share permission, but never tested that so cannot confirm if it works or not.

    I would consider first backing up to local disk then using something like robocopy to copy the files to the network share, for added safety and doesnt stop the backup completing if there is a network blip.