Using Chrome on Windows, I can right-click anywhere in the browser window, select "LastPass", select "Copy Password", and click the domain/user. (There's usually only one listed, unless I have multiple accounts on the site.)

Then just click the password field and Ctrl-V, of course.

If you're concerned about the clipboard, get in the habit of typing Shift-LeftArrow Ctrl-C after entering the password, to quickly replace your clipboard with a single character.