I have LastPass with binary component installed and copying passwords doesn't work on Ubuntu 16.04 LTS with LP support confirming the issue without ETA.
Lastpass icon changes to show you have an entry. Click the icon, click "Show matched sites." Hover over the entry and you'll see three buttons: Copy username, Copy Password and More options.
I'm assuming these buttons don't work for you. Right-click instead on the entry and you should see a menu with the following items: Autofill, Copy username, Copy password, Copy URL, Go to URL, Edit, Delete.
Try the copy password entry. Hope it works for you.