I'd forgo wood completely, and start with a base made from 1" to 2" box aluminium tubing. This also becomes the chassis to hold the draw-bar and the axles.
If you can tig-weld it that would be lightest and strongest, but bolting it together with corner brackets would work. If you use bolts also make and fit crush tubes inside the box-sections.
For a solid camper, I'd aim to keep it as low as possible else towing it will be a lot of work for the rider. Tall enough to sit up inside would be a fair compromise - give up on standing inside because it will act like a sail.
Upright members could be 1" aluminium angle, and again welded but also rivets or bolts are more workable too.
As a cladding, coroplast works well and even offers a degree of insulation while remaining lightweight. Mostly in white helps reject insolation too.
Tent material works as a cladding also, and can be removed and bundled up for the ride. However its not warm so only suitable for summer.
Why is weight so important here?
The average cyclist will produce around 100-200W of motive power when riding, or around a quarter of a horsepower. If everything weights 100-150 kg that's a Power-to-Weight ratio of 2/3 to 2W/kg.
The Caravan and Motorhome Club recommends that you aim for a minimum of 40 hp/tonne.
A car towing a caravan would be around 2000 kg each, for 4 tonnes, or 160 horse power. That's well within the capacity of most modern cars.
Apply the same ratio to a 150 kg load would suggest a power requirement of 6.7 horsepower, or around 25 cyclists.
Clearly bicycle-towed campers have enormous weight restrictions that don't apply to car-towed campers, hence the need for lightness.
Extras for experts - make the camper smaller for towing but extendable for camping. You could use a pop-top design where there's a roof that goes up on tent poles like this "Kamp-rite midget bushtrekka or another technology could be an inflatable like these examples The upside is easier to tow, but more setup required on-site.
Or just load a regular tent into a general-purpose bike trailer.