I am assuming that the door is smooth. If so, you can make it double as a wide ironing surface by adding foil and some layers of canvas or white denim over that. I think I have 5 layers, over foil and plywood. Just place them over - their weight holds them in place and you can peel a layer off to wash.
You can use a JoAnn coupon to get a fat reduction on a single cut of fabric, thereby saving a lot, Use very heavy foil.
That way there will be no cushion or wiggle under your cutting mat, yet you can still iron, too.
hugs,
charlotte