I can't address you problems directly...my growing conditions are vastly different than yours. Just a thought...can you heavily mulch the in-ground trees, even to the point of spreading cheap, plastic drop cloths over the mulch/ground to conserve moisture, rather than taking a chance on watering the trees and waterlogging thr figs as they ripen? In this way, the trees will use available moisture at the roots, and not get excessively wet.
I grow in large containers, in NYC. When that "time of month" comes...the containers get covered with plastic "caps" so that rain cannot soak into the growing medium as the figs get ripe. It does make a difference in sugar, and flavor concentration. Like the difference between a grape, and a raisin. And I try to wait until the figs just start to shrivel slightly before I pick them. Then they are thick with honey-figgy flavors. Oy!...I could plotz!
If any of the above doesn't apply to your situation, just forget the posting.
Frank