I've developed a Flutter plugin (sirikit_media_intents) that enables SiriKit Media Intents implementation into the app. If you take a look at the source code you can find how to forward iOS native intents handlers calls to the Flutter app.

I've used the in-app intent handling approach (iOS 13+), which doesn't require any additional UI or app extension. On the other hand, handling SiriKit intents in the app requires your app to enable multiple scenes support. So, this requires some adjustments to the iOS generated code of your app.