Sorry for the trick of perspective.

Here's a front view:

As you can see, I don't have cardboard directly over the sensor.
I just read the instruction manual again, and it said outright that glare coming off the bullet plays tricks on the sensors when it comes to shiny bullets and bright light. As far as the general impact of direct, non-diffused sunlight, it was less clear. It seems, from reading about how it works, that a steep angle for the sun and non-diffused light might get enough "intense" light into the censor recesses to mask the light changes created by the bullets passing overhead, but that is my guess, not something the manufacturer said explicitly.
As far as the "clear view of the sky" goes, that's stated in the recommendations for bright days when the chrono is placed in shadow. The purpose of the "clear view of the sky" seems to be to make sure that's there's adequate light for sensor function. I don't believe that an obscured view of the sky in and of itself would be a problem as long as their was sufficient light otherwise.
I've been thinking about a doghouse since 1SOW mentioned it last night. I'm confident that's the way to go for consistent readings from one day to the next. If I build one, and the Magic 8-Ball says "signs point to yes," I'll post pics of it.

I'll try to make it prettier than the above cardboard blind.