Illumination
In Strickland's paper, it's mentioned that they used 8000 mcd LEDs from [http://www.theledlight.com theledlight.com], with 20º viewing angle and 468 nm at 3.4 V.
It's also mentioned that a single LED was used per tube, with a cross section of 0.12 cm2 = 1.2x10-5 m2.
20º viewing angle, 0.35 rad, means a solid angle of 2π(1-cos(0.35)) = 0.381 sr.
This gives a luminous flux of 0.381*8 = 3.05 lm per LED.
The illuminance they used was then 3.05/1.2x10-5 = 2.54x105 lux (or lm/ m2).
A bioreactor that has some 0.2*π*0.3 = 0.19 m2, we would need at least 2.54x105 * 0.19 = 48260 lm, or some 16000 LEDs just to make sure the outer surface of the bioreactor has the same light conditions!!
This doesn't take into account reflections in the bioreactor, so I hope it will lower this number.
Illuminance test
We can try first different illuminance values, to get an idea about the range in which the LovTAP-VP16 switch saturates. This could be done by using cylindrical cell culture test plates (there are two sizes: 8.96 sqcm and 3.60 sqcm of circular base).
If we take the one with 3.60 sqcm (or 2.14 cm in diameter), considering we have 20º LEDs, the distance between the well bottom and the LED has to be of around 6 cm to have the whole well illuminated.