No, unfortunately not. I asked an Apple WebKit engineer and I interpreted the vague answer that they have some security features build into WKWebView, which basically prevent you to run a self-built copy.
I was able to solve my initial problem of using getDisplayMedia() in older macOS versions (it's officially supported in macOS Ventura) with a private WebKit API. Service Workers actually work in WKWebView since a while. I didn't figure out how to use WebInspector directly in a WKWebView, you need to run your app in Xcode and connect to it using the Develop menu in Safari.
dansch
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