Once I got the thing rigged up, it reminded me of the famous Surrealist motto, "A chance encounter between an umbrella and a sewing machine on a dissecting table." I decided to name my invention in honor of Isidore-Lucien Ducasse, the original author of the phrase.
Close up of the rigging:
Testing with the pochade box attached. The sun is low; you can see how the box is in shade and the shadow of the umbrella on the wall behind. It was pretty easy to adjust the wheel and handle bars to shade the box area.
The next day I tried it with the addition of guy ropes to stabilize the bike. I also used a higher quality umbrella (BestBrella). Instead of clamp or slipknots to hold the reins, I used the same nifty quick-release fittings (Nite Ize) as the guy ropes. Unfortunately, when the sun is higher in the sky the angles don't work out so well. I couldn't find a good setting for the wheel and handlebar to shade the box. However, the umbrella could clamp onto the bike almost anywhere, sticking out at almost any angle, and the guy ropes would stop the bike from being pulled over.
So, it turned out that the guy ropes were the most useful thing to come out of this exercise. I haven't used used the wheel and handlebar setup since, because the BestBrella's heavy duty clamp combined with the guy ropes' stabilization covers most situations. Below is the setup I use now. The guy lines are very useful in windy conditions or when the ground is soft. The reins are not usually needed because the umbrella is not attached to the wheel or handlebars.
I'm still glad I tried out the wheel idea, if only for that first photo.