I'm wondering about the rational for four wheels. I would expect this to be much harder to steer, and to be more affected by the rough stuff you're likely to encounter. Soft sand is also likely to be a problem. Admittedly I haven't done a lot of pushing of four wheels, but I have done a couple of thousand km of desert cycling in Australia and Mexico, (of which maybe fifty km involved pushing)
Any cart I have seen for desert travel has always been a two wheeler, like a sulky or gig. The options on wheels are for MTB or motorbike wheels or smaller diameter balloon tyres
Shows the cart Warren and Charles used, pulling 250kg over about 500km of mostly sand. My experience on sand is that it's all about the flotation, and big tyre bags and low inlflation are crucial.
The advantage of the Chinese wheelbarrow is that only a single wheel track is required, and that it is designed to take that sort of weight. I'm assuming about fifty to sixty days. If you're thinking non-resupply, you're looking at closer to five hundred kg(1000 lb) with food and gear. If you know someone with a spare wheel for a DC-10 I reckon you might be in business!! Relatively light, balloon tyre with a flexible casing that will take the weight easily. The other advantage with this is that you have one of you in harness out the front, and one on the shafts at the back pushing and stabilizing. It lets you mix it up a bit and steer around and over obstructions more easily. The wheel takes 100% of the weight.
A modern example of this is the MRT stretcher wheel, that allows you to carry a 140kg patient with two people over extended rough terrain. Not sure about the 300 kg limit though, or the flotation over sand. You might consider taking one each, which would give you good maneuverability over difficult terrain, allow you to double up for the really rough stuff, and give you some good redundancy if required. Less of a team effort though.
As far as shelter goes, I strongly agree with Nick's post in the other thread to avoid zippers. Fifty days use in sand will kill them. A shaped tarp is probably a great way to go for minimal faffing around. Maybe an oversized trailstar, since it doesn't have a zip, but you can pitch it all the way to the ground if required. A solid inner can be very nice when the winds get up, but if you have a valance you may get away without. Something like an MSR Twin Sisters or even Twin Brothers might also work.
Keep in mind that if there is no shade, the desert floor will be VERY hot, even when you put your shelter up at midday. If there is anyway to design the cart that you can rig the shelter over it, and use it to as a sleep platform for a siesta, you will be much more comfortable. Even getting a foot or eighteen inches off the ground will help. Also check the expected temps at night, and pack accordingly. I remember riding in the predawn with socks on my hands, as it never occurred to me to take warm gloves. A couple of pairs of leather riggers gloves might be worth their weight for warmth, and working hot metal during the day.
For shoes, it sounds old school, but there's a reason they're called desert boots. The suede keeps the rubbish out better than mesh and stays wet with sweat for convective cooling. The sole is stitched on, so there's no glue to melt. And they're not too heavy.
A light coloured cotton or canvas shirt works well. I like to use something with a bit of body to it, as I think it holds the sweat better than a very lightweight one
I hope some of this might help, and I look forward to checking out the video when you're done. I expect to see a swim in there somewhere!