Coleus Issue

First of all, thank you for the excellent follow-up picture. It is quite helpful. Good luck with the plant.
This may be bacterial. That would normally occur if the plant were quite stressed. You have it under lights, but how far are the lights from the plant? They should be no more than 12-18″ above the foliage. Also, if the plant is quite dense, the air circulation may not be adequate which encourages all the diseases. Perhaps you could pluck all the diseased leaves, fertilize at 1/2 strength and cut back on the water. Coleus as a houseplant is a bit iffy at best and there is not a lot of good photosynthetic area on the really nicely variegated leaves. Warmth is also a factor. They are happiest at 50 degrees or more and start to have problems at 40.