BES - BlackBerry exception to policy Disable Third Party Software Downloads
One way around this type of thing that I have used in the past is to create a completely open policy for installing software (while the device is in possession of the service desk, generally), and then pushed all the software and did any other configuration before applying the lockdown policy. Also, make sure your users aren't ignoring the prompt to set a vault password; I've seen this cause the device to completely stop accepting any type of push from the BES for some reason.
The disable third party downloads setting is not supposed to prevent pushes from the BES ever; it's only meant to disable downloads from AppWorld. This strikes me as a bug.