We have just released Screens 3.3 for iOS. Aside from the usual stability improvements, we were finally able to improve our hardware keyboard support significantly.
With version 3.3, you no longer need to tap on the Keyboard button to use your Bluetooth keyboard. Just start typing and Screens will detect that you are using a physical keyboard!
This new version also fixes that bug where an external keyboard could not be used in Fullscreen mode (bottom toolbar hidden).
Update: We discovered an issue where the HUD windows could cause the keyboard to briefly stop responding. Screens 3.3.1, which was just submitted to Apple, will solve this issue. In the meantime, the recommended workaround is to simply keep taping the Keyboard button before typing.