If there's some malicious desktop program running on your machine at the same time as the password manager then you're probably screwed regardless of whether the password passes through the clipboard - but maybe there's some subtleties I'm missing here.
If I have to do it, I immediately copy something else right afterwards, to avoid an accidental paste later on.
Could be better if it auto-cleared after a paste, but not sure how feasible that would be.
Personally I use rofi-pass to access GNU pass passwords (so just a hotkey and search for password name)
I'd warmly recommend it (or whatever other client you use for GNU pass) but I'm sure you can find similar functionality for any other decent password manager.
Makes them easy to use, with no saved PW.
That would probably be the best of both, no clipboard, and no browser addons.
There are other URL in title extensions for most browsers. And they're simple enough you can audit them or write your own.