In order to make the die not cut those little spots, you would need to figure out a way to have your machine NOT put pressure on just those spots. If you try covering them with something, you're actually going to put more pressure on them.
You could get really fancy and use a shim that has those spots cut out of it, but then your stack might be too thick for your machine. You could even "alter" one of your clear cutting plates to have holes in those spots so that they don't cut, but that sounds like an awful lot of work to me
If you're cutting this out of chipboard or matboard and then covering it with paper, I'd cut the chipboard and then take those little cutout pieces and pop them back into their matching holes and then cover with the paper. I know from other item's I've cut from chipboard that the cutout pieces fit pretty tightly back into the holes they came out of - almost like snapping together puzzle pieces.