Is there anything that can be done, perhaps writing something custom (like google apps script) or configuration on my end so that on open the addon automatically starts for this document and this document only?
Based on my experience developing unlisted add-ons, at this time it's not possible for end-users to add features to add-ons, like automatically clicking a menu, only the add-on developer could add them. In this case the add-on developer could add a mean for the end-user to create an installable on open trigger.
AFAIK the above is not explicitly mentioned on https://developers.google.com/apps-script.
My suggestion is to send feedback to the add-on developer.