The alternative you suggest is great in theory. And it might even work. But chances are it will do more harm than good if attempted by a novice.
We have heard of a technique, similar to the one you suggest, that is used by some contractors to raise large masses of concrete. Here, air bladders are placed below the structure to be raised, the structure is temporarily supported, and then grout is pumped (under high pressure) into the void as a permanent support. The process requires heavy equipment, is extremely expensive, and would probably cost more than the price to remove and replace the fireplace.
The chance of successfully moving the fireplace back against the house would be chancy at best, and could cause further damage to both the house and fireplace. This would be especially true if a solid concrete foundation did not exist below the tilted fireplace.
We still think stabilization of the slightly tilted fireplace is the most sensible and cost-effective way to get the job done. But thanks again for your comment.