Easy build, high weight capacity trailer for Burning Man
It's not too hard to build a trailer that will handle 300lbs / 150kg, but moving the trailer and staying on the bike in front of it are very different problems.

First, moving the weight. The playa is not smooth, so you're going to need more force than you might expect just to make it move. But since you'll be going slowly you can simply gear down to deal with that. It will help if you have bigger wheels and fatter tyres, so the 24" wheels would IMO be a better option. Those are not as strong when dealing with side loads, but again you're not going fast so the big problem will be people jumping on the trailer rather than fast cornering. Try to load the trailer so it's not inviting to climb onto.

Second, staying on the bike. The simple part is balancing at low speed. If your bike is geared low enough that should be doable, I'd put the seat down so you can put feet on the ground easily and stand to pedal if you need to. The more difficult part is when one wheel hits a bump or pothole and the 150kg trailer decides to throw you sideways. The trailer might also be able to lift your rear wheel off the ground, at which point you're going to fall over. The other obvious route to the ground is someone stepping onto the back of trailer. You can reduce that problem by putting a bit of weight on the back of the bike - maybe put your battery pack there. That will also help with traction, especially if you're standing up to pedal.

If you can mount to one axle of the bike you'll be more stable. The higher up the attachment point on the bike the better leverage the trailer has to push you sideways. Ideally the hitch would be below the axle, and with a 26" wheel mountain bike doing the towing and 24" wheels on the trailer that will also help make the trailer level. Even dropping it an inch or two will help. 

Hitch

If you can get access to a welder/someone to weld for you, that spring-and-chain hitch is very robust and unlikely to break or fail. It's pretty easy to make and if you prep it there's only 4 or 5 small welds to complete it. 

But for a week of BM you can probably get away with just about anything. Even just cutting up a truck inner tube to make giant rubber bands and using them to tie a flexible lashing on the rear rack would probably work (test it well before you go, because if it doesn't work on site you're going to be grumpy).

Wheels

I think you underestimate the cheap wheels found on kids bikes. For a couple of weeks with 100kg on them, most 20" wheels will do fine. And kids bikes are often thrown away, so you can probably get them very cheaply. You will pay more for decent tyres (or a supply of puncture repair kits and tubes, if you decide to spend your time fixing punctures instead).

I think the main issue you're going to have is the dropout area. You need to distribute the axle load along the bamboo somehow, so it might be worth using MTB front forks on the wheels (running horizontally) as the attachment points for your lashing. If you use "recycled" kids bikes just cut the forks off at the fork crown, or even leave the steerer tube in place. That way you have a fairly long section to lash to (albeit all the load is at one end.

Design

I suggest a basic design like my megatrailer, scaled to fit your requirements. You could lash that up with bamboo, the main addition will be cross bracing of the bed so it doesn't turn into a non-rectangular parallelogram (wire or rope would also work). I've seen this rough design made of lashed bamboo before and it seemed to work.



A chariot design would also work - make the towing arm 2-4 metres long to reduce the forces at the bike end of it. That would simplify everything except getting it to the event. The more I think about that idea the more I like it. You can just use 2-3 lengths of thick bamboo as the towing arm, strap the front end to your bike rack and even just run one length down each side of the trailer. If you did that you'd want 4-6m lengths of bamboo, but it would look quite cool IMO.