I believe that third party browser engines will have the same privileges as WebKit (in the EU, on iOS only).
PWAs do not run directly on WebKit; they run on Web.app which is a WebKit-based application. Think something like Cordova, provided by the OS, but which does not require each PWA to be a separate wrapped app listed in the App Store.
There is a recently-added interface to create one of these Web.app PWAs by third party browsers, as well as to create home screen bookmarks (which I believe open in the default browser not the running browser, but I am not a browser implementer).
There is not however not a way for a browser vendor to ship their own Web.app-equivalent using their browser engine.
Simulating some of the behaviors of a PWA on iOS outside of Web.app is also difficult. For instance - the multi-window functionality of iPadOS is not present, so a fullscreen web app would block usage of other tabs.