The Color Checker chart (regular or mini) will not work in making a camera profile. The 16 patches on this chart are not enough to make a good representation of the color space that is being measured. Also, the i1 Match software does not support this chart, but only supports the GretagMacbeth ColorChecker SG.
I believe the Mini ColorChecker is included in the i1 Photo kit so that photographers can use it in a scene to gray-balance the camera, and then confirm the gray balance during post processing in Photoshop.