I mentally figured out a workaround for updating while saving progress, but I haven't tried it yet. Plug your phone into a computer and locate the
folder when browsing the file system on your phone using your computer. Move this folder to your computer and uninstall the game from your phone. Install the game again using the APK downloaded from here, then move the folder back to where you found it. I have successfully done this to transfer save data between the version from the Google Play Store and the version from this website. I don't know however if this workaround works for the issue of updating the game.