Mikal - please do contact Valve via the options described here to describe your situation and use case so Valve is aware. Perhaps they could at least add a fallback option for that case.
It would be worth checking on the Steam Deck as well though - as far as I'm aware the overlay is working fine there, because it uses a simpler method of just drawing on top of the app, instead of intrusively modifying what the app renders.
Without overlay on, display webview2 window, only. When the overlay is on...
With any workaround that depends on only turning on when the overlay shown/hidden events fire, you still lose things like the "achievement unlocked" toast popup, and anything else that shows outside of the full overlay being shown. So any such workarounds will still not completely support all aspects of the in-game overlay.