Both of the photos you included show Compressor settings for an MPEG-2 Elementary Stream which is typically only used in DVD authoring. I don't know which version of Compressor you have, but since your client requires an H.264 file, look for an H.264 preset in Compressor - there should be a few of them. Once you've found a preset, duplicate it - the duplicate will be in the Custom Preset folder. Select the duplicate preset then make the changes you need in the Inspector window so it matches their requirements. Use the new preset to encode the video.
-DH