I'd guess this scenario results is a Cold War full of espionage. Direct warfare would escalate into nuclear bombing by Black.

So White should develop a horrible doomsday device to counter the nuclear bomb threat. Like redirect the edge-soulfire into all of the world. Then some controllers take an oath when and how to do that. Let Black read their minds. Oaths + mind reading is a perfect pre-commitment device. This essentially nullifies Black's direct attack power. It also means Black must work around arbitrary limits for winning, like "We destroy the world, if White leader is assassinated".

If White is unable to create a doomsday device, maybe they can fake it? Since you need to destroy the world anyways, this might simply the doomsday device activating, though.

You are not clear about the goals of White. World peace I assume. So the final goal is get all Black leaders take an oath you control. Thus, White needs to capture them, bring them to the altar, and force them into an oath for world peace. Luckily, Oaths are really great for espionage.

For one, you can perfectly turn any enemy spies once you capture them. Combine with memory erasure spells to hide the turning from Black. Is memory erasure the same as soul destruction? Then accumulating memories/experience means your soul grows?

For White's own spies them same procedure is great. Make them take an oath, the erase the memory of the oath from their minds. Perfect sleeper agents.

How would Black counter this spy superpowers? Probably lots of horrible tortures and tests to check for various unknown oaths. For example, immigrants must meet with illusions of the leaders to reveal any sleeper assassins who awake once near a Black leader. All naive direct attacks can probably detected this way. This means White must come up with a convoluted complex (cinematic) plan to capture leaders. Combined with the problems of exfiltration, this would make for a nice story in itself.