My suggestions are theoretical only; I have not tried them myself, but this is what I would try if faced with this situation. I would try to put a dab of glue on the corners of the oversized fabric, set it on the waxed side of the freezer paper, let it dry a bit, then iron. It might still let go except where glued, but at least it should lie flat beneath the signature fabric.
Another thing I think I would probably do, given the handling and traveling, etc. is to sew the very edges of the once-again oversized fabric to a piece of cardstock. Then you absolutely know it's going to stay in place. This way, there's no possibility of glue residue and you can use up some of your dull needles and rotary blades to cut the signed piece away from the cardstock when you return back home.
Hope these suggestions help. Let us know how you do.
And, finally, blessings for such a thoughtful gift.
~Jo