most times if you uninstall the app, it doesn't erase the game's log in information, you can simply reinstall with the new mod apk and it should be good to go. If you are having difficulties maintaining the login credentials, try to not log into google play when the window first opens, i.e. click outside of it and let the game open without logging into Gplay. that way you should have a local save of your file which you can then easily keep through installs and uninstalls. The other thing you can do is download the app called Titanium Backup... get the modded APK of it online (easy to find). It allows you to make the most comprehensive backups of apps data and install files. Try making backups of the game's information through Titanium with regular game install, then uninstall the game, then install the modded APK, then restore the games data.
These are just recommendations. I am not a wizard nor do I know everything about how all this works. Just my best recommendations for your guys that are having troubles.