First, I'm not an engineer, fabricator, nor welder. But I've watched lots of things built, have asked lots of questions over the years, and try to use as much common sense as possible.
There are two types of loads: static and dynamic. If you were cantalevering the bed out beyond the box for a stationary truck, you'd have only the weight of the bed area to worry about. But since the truck will go down the road, possibly with people in the bed or with stuff stored up there, you need to be concerned about the dynamic loads ... up and down as well as lateral (right to left).
I don't think the problem is a single big bounce going down the road with two 300 pounders laying up against the front of the bed (maximum leverage), but the fatigue over time.
I presume that the top of the bed box will level with the top of the mainbox. If so, start out with the top rail of the bed box extending back across the top of the upright studs of the box as far back as you can go (i.e. get long steel tubing). That will distribute the loads back into the main box before the first weld or even without a weld (all the way to the back of the main box). This will reduce issues with fatigue on a weld in shear because the tube that weight is suspended from will sit on the studs.
I don't know if 2x2 tubing is big enough or not. If you can find it and afford it, consider 2x3 or 2x4 (longer part on the sides, not on the top and bottom; but you probably knew that already). I think 0.125 (1/8") should be enough. Consider talking to a local racer with a late model to find out what their chassis are made of. Many of them are 0.125, sometimes a little heavier.
As suggested above, put in a triangular tube. I think this should go from the top, front corner of the bed box to the lower, back corner (up against the main box); I think the engineering logic to to squeeze the cross piece, not stretch it. That should give you the best strength.
I recommend that you have a triangular bar on the top and bottom of the bed box, or an X. I would gusset all the corners on the bed box with 1/4" triangular plates and I would gusset all the joints between the long top plates and the studs and studs and the bottom plates. Also consider X'ing the roof from side to side a few times.
Like I said, not a welder, but everything I've ever read or been told about welding and getting good strong joints is to make sure you chamfer the ends of the tubes to help ensure you get good penetration and good strong welds.
Finally, consider putting in some inspection access holes so that you can look at the critical welds; like where the bottom of the bed box attaches to the big box and the front stud of the bed box and the top plate.
In the bed box, consider running in 110v and 12v to receptacles on either side and even the front if deep enough. Little reading lights would be nice too. Because hot air rises, consider a small fan or two and even windows. Who likes to hear a noise in the night and you can't even look outside.
I hope I've helped or at least given you somethings to think about. Good luck with the project and keep us posted.