In the render dialog check "multichannel mapping" and open the channel selector and pick the surround bus.
In render setup dialog set the number of channels to 6. You will get 6 mono channels in the DNxHD render.
Note, you can't use this to make a surround H.264/MP4 file in Handbrake. HB will see it as 6 different audio streams and allow you to pick only one of the 6 channels. The only way I've found to make a surround MP4 in HB is to render to an uncompressed AVI in Vegas. That works but you need a lot of disk space for the AVI. A lot.
I'm worried that the channels might go to the incorrect speaker channel on their end.
Would I be better off with AC3?
Would that entail using a third party program to mux AC3 into the Avid DNxHD codec?