SignTool error when running build through Jenkins as another user
It's hard to say for sure, but if I'm understanding the question correctly, I'd suggest focusing your efforts on any environment variables or other user-specific config that gets passed from the parent process (scheduled task) when it spawns the child process (Tomcat). Even though Tomcat is running as Admin, it's possible that %USERNAME% is being passed to it as John.admin, or perhaps %USERPROFILE% or even %PATH% is being passed in incorrectly.
Another possibility: for scheduled tasks in particular, I've been burned in the past by not setting the "start in" folder properly, even when the program/script can be found.
To troubleshoot, perhaps write the values of your environment variables, current directory, and other relevant config info to a log file by temporarily modifying startup.bat.
I'm not terribly familiar with SignTool, but this might also help --SignTool Error: ISignedCode::Sign returned error: 0x80092006 -- the answers discuss where the private key is being stored vs. searched for.