Is this still and issue or have you managed to solve it?
When you say the email is not inter-company, could it be the the non-company email is another POP3 setup which has a rule to forward to his Blackberry account?
If it is a standard Exchange Server with BES installed and configured then the only reason why email will flow to the bbery only if Outlook is open is due to the fact that the BES server may not be setup for wireless syn on the second domaain.
Is it only one user that is having this issue?
Can you provide more details. I have installed and configured several Blackberry Servers and may be of assistance.