Hi Sharon -- I save the brown paper that's used as packing material for this exact purpose, but instead of using it exclusively as the bottom layer, I use it like I do the newspaper, to patch the uncovered areas, corners, and edges.
I wonder about your cardboard concerns, though. IMO, the main key to successfully engaging the soil underneath the lasagna is what you use along with and on top of the cardboard. If you have enough soil/manure/compost/mulch -- and give it enough time -- it should work fine. (Though in more compacted spots it can take a while.)
In other words, yes to the kraft paper but I don't think the cardboard is the cause of the problem you refer to.
I just did a much larger lasagna project, and am scrambling to get enough stuff on top! And I did have to patch among the cardboard pieces with every bit of paper I could lay my hands on, to complete the shape.
