If you want to use Edge in OBS, you can capture an Edge window. But you cannot exchange the browser library within OBS.
What Edge browser feature are you interested in that's missing in the integrated browser? After all, internally all 3 browsers are based on Chromium: Edge is nothing more than a rebranded Chrome with some Google service pointing libraries exchanged with Microsoft service pointing libraries.
Keep in mind the integrated browser is designed to mainly render and show the content of websites. It's not designed for much interacting with websites. Keyboard (for Chat) in browser docks works, but that's about all. So if you need a full browser, capture the window of a full browser.