I wonder if this will require a microprocessor like Shine boards.
The only thing I have to say is that you might not want spherical caps, they're generally less natural to type on than sculptured. However, if the case is extremely tall, that might be your only option. But it wouldn't matter if it stuck out or not, because it's in bright lighting, correct?
Spherical/cylindrical and sculpted/flat are independant quantities.
For example, TG3 uses a flat profile cylindrical keycaps (same profile as SP's function row) on their BL82. Most keyboards are cylindrical and sculpted.
I am currently typing this on a set of retro DSA keycaps that are flat profile and spherical. Many vintage keyboards (like that on my selectric II) are sculpted profile spherical keycaps.
Sculpted really just means they use different profiles for the rows to better "sculpt" it for typing.
As far as microprocessors, Not sure what you mean here: most keyboards have some sort of microcontroller controlling them and converting keystrokes to scancodes. Microcontrollers can be thought of as similar to processors, though the word "processor" has arguably changed throughout the history of computing.
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@OP: Blues are an excellent choice, though you may want to consider a spring swap to make them heavier. For some reason, heavier switches have more of an old-timey feeling that the light and clicky blues.