I am in the middle of making a "checkerboard" cake.... the last time I did this, I used a chocolate cake mix and a white cake mix....... I had a TERRIBLE time.... someone told me I couldn't do that and expect the cho and white to bake at the same time table... ( the cho didn't get done enough) that I needed to add cocoa to the white batter. Okay, I did that.... and the chocolate batter is SOOO thick, I don't think it is going to turn out. Should I have added more liquid to the chocolate batter ? It is in the oven as we speak....
I have made the checkerboard cake and used chocolate and white cake mix but I used the same brand. It turned out fine. I might be wrong but I think you would need a little more liquid for the cake mix where you added the cocoa. I hope your cake turns out for you.
I wish I could help you out on this one. I bought a checkerboard cake pan set and tried 3 times with terrible results. So I returned the pan.
Unfortunately the ONLY way to find out if it works is to cut the cake.
I personally wouldn't bother.
I know this is not helping you....I'm so sorry 
I've made two of them... the first time it was the candy corn cakes with great GREAT results... used white cake and colored it orange and yellow.
The second time I used white cake and colored it red and blue for 4th of July.
I'm thinking if you use either the same brand cake mixes, 1 white 1 chocolate it should turn out fine... have you tried using yellow instead of white? For some reason white acts differently then any other cake lol either rises too quick or not enough or takes too long or gets done too quick lol....
I would suggest maybe using yellow instead of white unless you HAVE to use white... and in that case.. I'd add cocoa powder just maybe 1/3 cup and a tsp or two of more water to thin it back down... don't add too much water though... add a little at a time until it's the right consistancy 
Good luck!
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