Height makes that decision a little easier, but how tall is too tall when you have a muddy, 85 pound wheel and tire to lift? You can toss the spare off the roof, but be careful where it will land. Are you near a dropoff? Mr. Murphy (and his law) will take over and ensure you tire rolls down that hill. You obviously don't want to slide it down the side of the vehicle because of the possibility of scratched paint and broken glass.
Rear door works well, but there are visibility concerns. Also, the tire might be too tall if you've gone up a few sizes and may not clear the bumper. You might be able to flip the tire carrier for a little more clearance. Stock size with a factory carrier on the door is ok. But are the hinges up for the task of a bigger tire and maybe a steel wheel, and a full Trasharoo?
Regardless of where the tire is mounted, if it's rattling, it's probably not secure. For a strap-it-down-on-the-roof setup (I do this for my classes when students will practice plugging a tire and breaking/reseating the bead), I use 2 ratchet straps rated to a WLL of 500 lbs. each.