We've been using double chamber compost toilets at our school for about 15 years. I'm sorry to report I don't have any great advice for your fly problem. Not because we tried hard and didn't succeed, but because we didn't really try very hard.
-- We did try the fly trap bottle thingie, putting the bottles on the upper outside of the manure chamber door, where they get direct sunlight for a few hours a day. They did collect flies, and did make a difference, but they didn't eliminate flies completely. The manure chamber doors have deformed over time and no longer close tight, so we don't even bother with the fly traps anymore. Anyway, the traps really bothered the Ladakhi Buddhists, the majority here, who have been preached to since childhood that you
should never kill any living creature, even a fly, and they seem to take it as "
especially not a fly."
-- Cover material and training the users to use it regularly makes the biggest difference, I think. We often use sawdust and
wood shavings from the
local lumberyards, and (whole) autumn leaves, but both of these are poor at preventing flies. When we're out of those, we use garden soil, and frankly that works better at reducing flies. The carbonaceous materials do well at controlling smell and I think they make better and lighter compost, which is important too, but the larger wood shavings and whole leaves allow flies access to the latest
poop. Fine sawdust would be better but we can't get much.
-- Our upstairs user rooms don't have a roof at all, which means it's not oppressively unpleasant even if there are some flies, and there is rarely an oppressive smell.
-- Our manure chambers are very big, 3 x 8 feet (1 x 2.5 m), which is great for handing the large numbers of people we have here (holding
enough for a whole year on each side), and wide enough to be convenient to empty. We average about 20 resident users per user-room, year round, and almost nobody pees elsewhere. We have two holes over each manure chamber (covering off the two holes over this year's unused chamber that is standing to compost). But I've also thought of a design, for a smaller population, where you'd have a much narrower manure chamber, so that when you throw the cover material down the hole after use, it forms a thick layer over the recent contribution, instead of falling far away to the sides. That might help with the fly problem, too. I think.
-- Aside from the bottle-light trap, which simply hopes that flies go in but can't figure out how to get out, there is the sticky-trap tactic. Nowadays, instead of the hanging spiral fly strips we used to use in the US, you can get "clear window fly traps." I use these in my residential quarters, and they are very effective, and much easier to handle than the coils. They are about 2 x 6 inches of clear plastic, sticky on one side with a narrow strip on the other side to stick to the glass. Especially in cold weather when the window is sunny and warm, the flies bump up against the window until they get stuck.
-- I've been thinking that introducing compost worms to the manure chamber might help, since maybe they would fill the ecological niche that the flies are using, without an annoying adult phase. But my colleagues are skeptical about that suggestion because of the fear of killing a bug. They say nobody will be willing to empty the chambers if the manure is full of worms. I said I heard that when we're not using the chamber for a year, the worms will migrate through holes in the wall to the fresh chamber. So this summer we got some worms and threw them in, but I won't have much to report for another year until we empty it. It's in the boy's toilet block so I haven't seen for myself if it reduced flies.
-- Oh, I forgot, we introduced urinals and a big ol' tank for
urine this summer in the boys block, but I haven't asked if that has reduced flies. The tank gets drained into the
irrigation canal when the
trees near the boy's toilet block are watered. Also I'm curious when we remove the manure after two years, if the cover materials have decomposed as usual or instead enclose desiccated turds. In this arid climate, that seems a possible risk to me.