I've likewise been plagued recently with sour fruit on my brown Turkey, perhaps due in part to rain, but mainly to the tiny beetles that enter through the open eye. I've heard them called "sour fig beetles" as well as "dried fruit beetles." The one in the attached photo has just exited the fig; it then flew away.
In an attempt to foil them, I tried sealing the ends of green figs with a drop of white glue, hoping to mimic the strategy of the varieties that seal the eye with "honey" or resin. It's sometimes effective, but seems to lower the quality of the fig, and I doubt I'll bother trying it again unless I can find a material that works better. The attached shot is of a ripe BT fig that was sealed a couple of weeks ago when it was green; it actually tasted pretty good, but not great. Has anyone tried sealing fig eyes with another substance that works better?
The other bad pest, that seems somehow associated with the little beetles are the big green June beetles (see attached) that eat the fruit from the outside in, leaving just a few shreds of dried skin. Both types of beetles are all over my brown Turkey, but I've never seen either on the black mission, just over a hundred feet away.