Orion creates profiles by making a proxy app that just calls the main app with an argument. It would be useful if this proxy app identified itself as a browser to the system. This would make it possible for the user to select a certain profile as the system default browser in the System Settings and for browser picker apps (for example, Velja) to be able to show Orion profiles (proxy apps).
To resolve this, all you have to do is to add the following to the Info.plist of the generated proxy app: https://github.com/sindresorhus/Safari-Private/blob/a1b08e15e38d685282da35c6e9fcdb14777b36b2/Safari%20Private/Info.plist#L5-L33