After a very interesting conversation with AI, I am now considering the following system.
Aircraft are allocated to theaters, as now. For example, the Eastern Front is the Luftwaffe against the Soviet Air Force.
There is a dogfight phase. Only air superiority aircraft participate in dogfights. This means fighters, advanced fighters, jets, and (new unit type!) heavy fighters. During the dogfight phase, all aircraft fire at their dogfight values. Destroyed aircraft are removed.
Then, there is an interception phase. Fighter-type aircraft fire at their interception values. Bombers fire back at their defensive fire values. Destroyed aircraft are removed. Damaged bombers are aborted.
Any bombers which weren't destroyed, and which weren't aborted get to bomb whatever targets they had in mind. Strafing armies, attacking navies, or strategic bombing raids.
To make this more concrete: a fighter would have a dogfight value of 3 and an interception value of 3. A heavy fighter would have a dogfight value of 2 and an interception value of 4 or maybe 5. Also it would cost a little more than a normal fighter.
I don't know for sure how this will actually work without play testing. But here is what I hope would happen. The Soviet player sees that Germany is able to allocate, say, 10 fighters and 6 dive bombers to the eastern front. The Soviets have 5 fighters, a dive bomber, and a heavy fighter they could allocate. The dive bomber hides in Siberia so it won't be shot down during the interception phase. The rest of the Soviet air are sent to the eastern front. The Soviets lose more air than the Germans during the dogfight phase. But, at least the German dive bombers didn't get to fire during the dogfight phase, so that made it a bit more even. Most of the Soviet fighters survived the dogfight, earning the chance to intercept the German dive bombers. The chance to kill or abort some of the dive bombers helps balance out this air battle. This balancing out is what induced the Soviet player to participate in the air battle in the first place, instead of hiding his entire air force in Siberia.