Projection & Cut Outs

I am trying to make a cavitiy of a part I am making by doing a cut away of the part i need the geometry from. Can someone help me with an easier method of doing this? I want the red part to be able to slide into the box as shown in the pictures but am not sure how to get the cut away into the box from my parts so that the geometry is exact. I tried a merge cut out by importing the part which failed and I tried a cut out in assembly mode  under component operations and that failed as well. Essentially I am trying to do is what is in the picture of the 3D printed structure.