So far on mobile it’s been relatively manual, often copy from the laptop screen (I have a mac+iphone so sometimes I copy&paste via the notes app). I don’t log in so often on mobile though, or better, I almost never log out.
But you nailed the issue — this is why we’re thinking to build an iPhone app first ( using react-native, so Android shouldn’t be too far after), and a Chrome extension after.
Secure/hardened implementation will come along the way, I think we’ll start with an alpha with just the basic functionality, assuming that your phone is secure enough in your hands. This said the custom keyboard is an interesting idea.