IMO, the lighting is also another contributing factor. Is 12 hrs required? Are you having highly photosynthetic corals that require long lightings? If not, still can reduce. I am having an average of 8 hrs lighting per day, 4hrs in the morning and 4 hours in the evening (for my viewing pleasure after work) with soft corals/LPS.
To get rid of the diatoms, apart from the silicate, try to off the light for at least 3-5 days to starve out the light source.
If your hand cannot carry heavy load, spread out the purchase of the distill water from supermarket. I used to get like 4 bottles each visit (2 bottles per hand).
Cheers and dun give up.