I am afraid private listings for Forge apps are not yet supported and are not currently prioritised. Distribution through the console is disabled for listed (or in the process of being listed) apps to prevent customers from installing an app without understanding and accepting its pricing.
While not ideal, have you considered registering a copy of the app and deploying it to production? As this will technically be a new unlisted app, you will be able to use the distribution link through the developer console. This won’t help with testing the licensing code though, but can help with dogfooding.
I understand this is not a great experience and will be keen to hear how critical are private listings for you and where do they sit (in terms of priority) among your other asks from the Forge team?