Umm ... by no stretch of the imagination could I call black soldier flies "little" !
If you want to cut down on the flies, cut down on the moisture. If you let your stuff in the tumbler dry out a little more, it may not compost quite as quickly, but it will be less hospitable to lots of different kinds of larvae. For right now, you can accomplish that just by throwing a bunch of extra browns (fall leaves, straw, shredded paper, etc) in and mixing them. They will absorb the excess moisture. (You do know about balancing "greens" and "browns," right?)
I have never had a tumbler, but from all the people who have written in here about them, it seems like they are prone to excess moisture, not draining well enough and sometimes getting stinky from combination of excess moisture and not enough air circulation. You can help with all of these issues by putting more air/drainage holes in it and then being sure you do spin it frequently (and keeping your balance a bit heavier on browns than you might with a compost pile on the ground and un covered).