To put it bluntly there are two ways to deal with an insurgency: mitigate the underlying reasons for dissatisfaction with the current order, or murder every man, woman, and child who might at any point be a potential recruit.
This Japan is an authoritarian state, so they hypothetically could liberalize their rule in the JPS but I doubt it, for the reason that liberalizing would necessitate admitting that they have been doing something wrong, which the Japanese leadership will never allow.
The likelier, but more unpleasant (but what in this show is pleasant?) option is to go full on exterminationist on the black population of the JPS before the events of season 4, enlisting the help of the white population against the black population. Given American racism and the time period I doubt many whites in the JPS would care about the fate of their Black countrymen, and frankly many whites would be more accepting of Japanese rule.
So the most likely answer that would possibly succeed in maintaining Japanese rule involves causing even more bloodshed than had actually happened in the show. This probably has the effect of cozying up the JPS with the Reich and continuing good relations for at least a while longer.
They have to chose between Mainland Asia/Oceania or Japanese Pacific States. Their remaining resources don't allow them to fight a two-front Vietnam War-style conflicts.
Admiral Inokuchi did propose a "hearts and minds campaign" so they would not need to deal with insurgents. It was all too late by then as the Empire suffered too much loses and casaulties, not to mention the rise of the Black Communist Rebellion. Interestingly enough, the "hearts and minds campaign" drew parallels to what the U.S. tried to do in South Vietnam.