To make it more successful, perhaps the Russians do not launch the rather pointless Lake Naroch offensive and instead allow the reserves for that attack to be assigned to the SW Front, where Brusilov was training his infantry divisions in infiltration and shock tactics before the Germans were. This might allow the Russian attack to be supported well enough that they can stop the Linnsingen counterattack and keep the momentum rolling.
The failure of Evert to make progress, and the diversion of his troops into Brusilov's gains only confused and stuttered the Russian advance with too many men in one area. Evert should instead only make a demonstration and keep attention fixed rather than launch clumsy and lazy assaults.
Brusilov went about as far as he was able to, but if he could somehow get farther, and conquer Galicia, A-H might consider a separate peace.