If you are using Mac Catalyst, you could create a bundle (NSBundle) which has direct access to the AppKit lifecycle and APIs and load the bundle from Mac Catalyst (just be sure you only load the bundle when you are running on Mac Catalyst because it'll crash your app on iOS, in fact you should only add the bundle to the Mac Catalyst target). From the bundle you can call anything in AppKit (and make a menu bar item).
This is a widely used technique in Catalyst apps to implement missing functionality that the framework doesn't provide. No private APIs are involved. Apple doesn't like to recommend this but I'm pretty sure SwiftUI just shims over the same APIs (related to status items and such) that they seemingly don't want us to access directly anymore, for whatever reason. Maybe they think they are "abstracting" things away to make it simpler for us but often, for me, it feels like they are tying one hand behind my back.