Just a quick summary of the root cause of the problem: BPS Pro and Quarantine are working correctly on the other sites hosted on this host, but this particular website where Quarantine is not working correctly is hosted on a different server. I found several server configuration mistakes on this server that confirm my original suspicion that the root cause of the problem is with that server itself. This website needs to be moved/migrated from this “bad” server to a “good” server where BPS Pro and Quarantine are working correctly.
UPDATE:
Per your email response and questions about not being able/allowed to move/migrate this site.
You will need to ask the host to look at these things below. One of these things is the cause of the problem:
Suhosin is not configured correctly.
Zend Guard or Zend Optimizer is not configured correctly.
IonCube Loader is not configured correctly.
The PHP server itself is not configured or compiled correctly.