WebExtensions can set a custom user agent on the page through browser.devtools.inspectedWindow.reload({userAgent: "CustomizedUserAgent"}) and this user agent will persist reload and navigation until the DevTools toolbox is closed (it's not at the moment in Firefox, but is in Chrome, and this will be fixed in Bug 1706098).

The thing is that it's up to the extension panel to show this new user agent, as there's no place in DevTools where this information is displayed (except Responsive Design Mode, but even then, it would only show the user agent value that was set when RDM started).

It would be nice to have a way to highlight the fact that the user agent was changed by the extension.

As a first step we could display a notification in the NotificationBox when an extension is overriding the user agent. This would not require too much work and we won't have to think about a good design.
A better step would be to reflect the user agent somewhere. Maybe in the extension panel title? We could borrow a piece of design of proton and display a USER AGENT text below the title (same idea as when you have video playing in proton).
Or maybe it could be simpler, an icon or a badge on the extension title, with a tooltip to reveal the custom user agent.