I need to make a 1/2 sheet cake for Sunday, and they want it half carrot cake & half chocolate cake. Only problem I thought of was that I'll need to use half cream cheese frosting & half buttercream & they are 2 different colors. How can I remedy this??

I wouldn't do that. I would either do two separate squares or make them pick one icing. It probably can be/has been done, but I wouldn't want to do it. JMO

You could make two square cakes, ice one with the cream cheese icing and the other with the buttercream icing. Cut a cake board to fit between the two cakes and push the two together leaving the cake board between them. You said that each side is going to be a different color. Does this mean your customer wants each side decorated differently? Or is it because the cream cheese icing would be a different shade from the buttercream?
Sandy