I can see how that would improve flow inside Foundry for sure. After a little dig into the possibilities, I'm thinking VTT integration is a whole world of its own and probably beyond scope... Foundry module developers could do a much better job anyway! Dungeon Reader is designed to be a 'reader' app first and foremost, with just enough tooling to enable the GM to stay in the book, and maintain immersion. I love the idea of it helping you in the VTT context, but all I can suggest (though it pains me to say it!) is to have it sitting happily in that other tab... 🙏
EDIT: Not ruling out some kind of snipping tool though, to copy a map or image straight off the page to the clipboard. Fairly straightforward to build and useful with any VTT... might make some of those things you mentioned more streamlined?