Investigate using GtkGraphicsOffload
With WebKit starting to use GdkDmabufTexture
(I'm referring to this comment) it might be worthwhile to put the webview widget into a GtkGraphicsOffload to make webpages able to gain the benefits outlined in this blog post, in particular bypassing GTK's compositing when only the web page changes, and in particular when fullscreen.
Of course, it makes GTK use different codepaths to maintain the offloading infrastructure, so there might be reasons to not use it - my particular concerns would be stability and unespected slowdowns, though I have no concrete reasons in mind, that's just gut feeling.
I have not thought about using graphics offload in a browser too much, as we've built it around video content, but I don't see why it wouldn't be beneficial to a browser, too. And as all the work required is literally just adding a single widget, it's certainly worth trying.