What are you planning on doing with the images, assuming you could get the correct color? It may not matter to you. Why are you taking 3 different photos instead of just one with all 3 lamps on simultaneously? You do not have non-standard illuminants - you do not have illuminants at all. No one does - they are a mathematical concept. You have a light "source", not an "illuminant". A reasonable LAB can be gotten from a light source as long as the light source spans the wavelength range of the illuminant you're trying to map into. But you have to take a picture of a known standard, such as the x-rite Color Checker Chart with your light source, then develop a transform from your RGB values into the XYZ values for the illuminant you want, then from XYZ to LAB. That's if you want to do calibrated imaging. If you're just trying to find bouncing red balls against a green screen, then you don't need anywhere near this kind of precision - it would be way, way overkill. In that simple case, just using the book formulas with any illuminant might be fine.
Spectrophotometers use white leds or xenon flash bulbs. They're not D65 or C or D50. Ask yourself how they are able to produce LABs for D65 or C or D50 when they do not have that illuminant as their light source. How can one light source give LAB's for virtually any illuminant on the same instrument? Once you understand, you can answer that.
