Keep in mind that most all soybeans grown in the USA are genetically engineered before considereing using it on your organic lawn.
Corn meal has a lot of protein which converts to Nitrogen.

From what I understand, the preemergent qualities of corn meal may not make it the most desirable product.

Corn meal is not particularly high in protein compared with other sources. Soybean meal is much higher in protein than corn meal. Corn meal does have protein and can be used as a fertilizer, but it takes more of it to provide the same amount of nitrogen as soybean meal.

The pre emergent is corn gluten meal, not corn meal. Corn gluten meal is very high in protein and is a decent fertilizer as well as a pre emergent.

Can I put down corn meal for fungus and afterward (a few days) put down soy meal?
Yes you can. As mentioned above SBM has great nitrogen release. Corn meal has anti-fungal properties. I use them both and highly recommend them. Cracked corn can effectively substitute for corn meal.
Bill Hill