A simple proof of concept chrome extension which skips the standard copy route itself to avoid this: https://github.com/awalGarg/realcopy
Note: User rain1 from chat suggested testing it on https://thejh.net/misc/website-terminal-copy-paste where it didn’t work. That site uses CSS to hide the actual selected text instead. The extension above however shows what you copied so it might still work for you anyways.
Edit: Also note that it puts the copied text in a textarea (separate from webpage context) so you can edit something if you like. Now I am actually thinking of using it, but needs an FF port :P