GPO push install fails with error code 1603
1603 fatal errors usually are caused by one of two things: the account doing the installation can't write where it needs to, or the installation package is corrupted. Since you've ruled out #2 by doing an install with it, I'll assume it's #1.
When you use a computer-based software installation GPO, what you're doing is installing the software as the SYSTEM account. Some software installers misbehave and require that data be written to the installing user's profile. The SYSTEM account doesn't have a traditional profile like other user accounts do, which gives the illusion that required directories do not exist. This could explain why a logon script or manual install works - neither run as SYSTEM.
I'd check with your software vendor and make sure that this software can be deployed the way that you are trying, since all signs point to the software installer itself being the culprit.