On iOS 16, when Lockdown Mode is activated:
- Orion does not appear in the list of apps under "Configure Web Browsing", and
- There is no way to exclude websites from Lockdown Mode (as there is in Safari).
A quick search on https:/developer.apple.com doesn't reveal much exposed functionality related to Lockdown mode, so both of these may be impossible to implement in Lockdown Mode's current state.
That said, some apps do get listed in Lockdown Mode and can be manually excluded, so it may be possible (though it's unclear how) to get Orion on this list. ("Data Saver Mode" is similar in many ways to Lockdown Mode's effects on Safari, so excluding Orion, turning on Data Saver, but then allowing users to exclude certain sites from Data Saver would be largely equivalent to, but more functional than, the current situation.)