Permanently grant right to control another app to Apple Script App
A regular AppleScript saves properties and global variables (including variables in either run handler) in the script itself (the script file is modified).  This is how persistent properties have been implemented, but now this can run into accessibility permissions issues, since modifying an application will cause the system to see it as a different one - one that hasn't been given permission.

The solution is to code sign your application (you can use a self-issued certificate) or make the scripts in the bundle run-only.  You will lose persistent properties, but you can use some AppleScriptObjC to implement preferences via NSUserDefaults if you need to keep stuff between runs.