Do not duct your exhaust into your heater, that is not safe.
You will prevent carbon monoxide from escaping the pipe from the heater because there won't be enough room in the line for all of the air to escape.Too much air volume. Every duct/pipe has a capacity and the gas heater exhaust isn't rated for that heavy of a load.
What you should do is vent directly into the attic, above insulation level. What kind of climate do you live in? Generally, the volume of air in an attic is an adequate buffer to prevent mold from moisture and heat build-up.
Also, a 300CFM fan attached to a carbon filter, (2) 600W lights, and ducting will not even come close to keeping that thing cool IME. Go back to the drawing board here.
To me, the ideal setup for you would be to cut a whole in your ceiling to fit the ducting through (keep the circular piece for later repair when you decide to move), mount your fan in the attic that is sucking the air through your equipment, and do a short but straight duct run to ensure the exhaust isn't blowing directly onto anything (like insulation).
For example:
Carbon filter>>>>>>600W Hood>>>>>>>600W Hood>>>>>>>Fan>>>>>>>>Attic
Minimize the bends in your ducting as much as possible, and go with an 8 inch fan with 8 inch flanged hoods. I recommend the Radiant 8; fantastic reflectors.
What are the ambient temperatures in the room where this box will be located? That may change some of my analysis a little, but for the most part I think it will stay the same.
The more equipment (fan, ballast, etc) you can keep on the outside of the box, the better. Get a small shelf to put on the side of the box for your ballasts to sit on along with other essentials.
When considering your carbon filter, you will want to go with a lower rating than your fan, not a lot, but definitely lower. That is because your fan will not be attached directly to the filter, which will alter the CFM rating. You have the choice of ceiling mounting the filter or setting it on a cinder block or two about plant level. I prefer the latter because it is easier to deal with moving it.
One other consideration: intake. Passive intake will likely not produce enough air flow to keep a negative pressure on the box. Buy a blower fan to force air into the box on the opposite side of the carbon filter. Also have your oscillator blowing towards the filter. This will create the most efficient path for the air to travel.
If anyone wants to call me out, feel free.
I have done a setup like this before and it worked flawlessly with temps 4 inches under the reflector of 77 degrees, ambient temps of 72. If you provide me with your climate information/temperature information I can help you further evaluate your setup.
Hope that helps you some. Happy Growing!