Hey,
Both iOS and macOS versions of Orion already support optionally setting a separate search engine for private browsing.
Head over to settings → Search to get started. E.g.:

Dinketron I don't want to deal with logging into Kagi for every private window.
This is also easy to fix in the same location! All you need to do is click the icon to the left of the Search engine field when Kagi is selected. Enter your session link there and using Kagi on private browsing is seamless.