Is your code snippet your actual code? Lines 11 and 15 are at odds with each other over the type of binaryHeader.

I cannot see an immediate reason Qt might do this unless the proxy has some odd limitation.

What does the memory do while you are moving your 1.3GB payload?
Have you used Fiddler/Wireshark to see if data is actually moving on the net when the proxy is in play?
What type of proxy are you using? HTTP or SOCKS?
Is it HTTP or an HTTPS site? The proxying mechanism may be different.