That actually would be 100% AGAINST code. I strongly recommend against doing that. Introducing current into the bonding conductor, and make no mistake it will have current on it from the unbalanced load, is a recipe for disaster. Not only can it reek havoc on your GFCI's ability to determine the mA differences in bond and neutral required for it to function properly, any bonded metal part between the receptacle and the panel that you touch could potentially send current through you if you happen to be touching something with less resistance to ground than your service (this happens in older homes where they were bonded to water pipes that were later replaced with PVC/PEX, or services with high resistance grounding from an apprentice not burying the plate deep enough, etc).
While it's true that the ground and neutral are connected at the consumer's service box, the practical impossibility of current coming back through the neutral to that point, then heading BACK out through the bonding conductor is the main difference.
Point is, if you don't have a neutral, ground, and 2 opposite hots in that box, pull a new wire or get a different dryer.