@ectcetera if I remember correctly - Godot4 web export did solve webbrowser copy-paste, Godot 3 always was like that - you need your own solution. But idk about 3.5+ web export - maybe there it also fixed, idk.
@ectcetera maybe you can make it without popup - maybe bind https://developer.mozilla.org/en-US/docs/Web/API/Element/paste_event to your function in godot that will receive text - so it will work by just hotkey as usual. (it is still junk, correct solution would require rebuilding godot-export template with modify of some code)
Element: paste event - Web APIs | MDN
The paste event of the Clipboard API is fired when the user has initiated a "paste" action through the browser's user interface.