Kellar,
It's my turn for my ignorance of Exchange to be exposed. I am also ignorant of Postie. I would assume it uses SMTP. We use Lotus Notes. I use SMTP to send mail from web forms on my intranet as well as from several of my NMS (Network Management Systems) apps like WhatsUP, UPS, and environment monitors. In our case, as long as I am within the same "A" class IP, I can bounce email off of our SMTP server. From another class, I cannot. This, I understand is a 'best practice' since your SMTP would be exposed to the internet and you would not want spammers bouncing mail off your server.
I thought if you had another resource like an intranet server to run some tests on, it would confirm whether your email admin is blocking you. Can you not work with your email admin to get you the functionality you need?
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Give a man a fish and he will be back for more. Slap him with a fish and he will go away forever.