I think that using the tissue paper underneath is a really good idea. It will keep the fabric from slipping on the bottom so much, and I'd pin together the
entire "sandwich" and cut it out as one. Also, I'd start working on the folded pieces first, and then do the single layer ones. You have extra fabric to work with, which is good insurance, but I'd layout and cut as though I had just enough, in case something shifts and you find that something needs re-cutting.
Just go slowly, working with as small a section as you can, double fold things first (IMHO), using the tissue paper underlay and pin pattern tissue, fabric and bottom tissue together as one. Cut them out all together.
If you are going to be cutting out on a cutting table, don't let the excess hang over the edge, as gravity will cause the fabric to pull or shift; and even slight movement will distort the grain.