If my character's proportions are very different than the original human's proportions that the mesh deform cage was designed to contain, then I have to do so much tweaking to match the cage to my character. Is there a way to use my own mesh cage that already matches the character and copy the deformation weights from the BlenRig mesh deform cage to it?
Yes, Although you may end up needing to spend just as much time correcting the weights.
The data transfer modifier can copy vertex weights, but you will probably need to do some tweaking after and of the modifier settings to get things right. It will also need the blenrig cage to be made to at least roughly match your new cage, as you are going to want to use the "Nearest Vertex" setting.