There's no indication that death curses are powered by the soul. The soul in Dresden Files is a source of energy, true, but it's also considered to be generally immortal. Wizards and other supernaturally-powered mortals (explicitly including the Knights of the Blackened Denarius and the Knights of the Cross) anticipate that their deaths will result in their souls traveling to an expected afterlife.
When he describes magic and how it works, Dresden describes it in ways similar to the Force - magic is created by life. Magic is within every person, and is an integral part of the difference between a living person and a dead one. Wizards, sorcerers, kinetomancers, ectomancers, *-mancers, etc draw upon their own internal reserves of magic as well as the magic around them.
Harry had a notable experience with this while he was contained in an enormous, Hellfire-charged pentagram that cut off access to outside magic. He pulled as much power as he could into his internal reserves and used it as efficiently as he could. Once that charge and the magic that had already been in the area were gone, he was unable to use magic.
Hellfire, which Harry gained access to for a while, is an external power source. It could be used to supercharge his spells, primarily those which were destructive.
Soulfire is different. Soulfire allows Harry to use the energy which makes up his immortal soul to make his magic more than it was. His magical constructs are more real, his fire spells burn truer, links between Play Dough and pieces broken off of the can are stronger and more lasting, etc. Soulfire, however, is fueled by Harry's soul. The implication is that if he pours too much into spells, he will exhaust his soul.
This is an important, huge difference between Soulfire and a Death Curse. Wizards need a moment to prepare their Death Curse. They need to know they will die, and make a conscious decision to expend every bit of magic they have (leaving none to sustain their bodies). They also know or anticipate that their souls will not be destroyed by this process - they still anticipate an afterlife.
On the other side, Harry has repeatedly stated that he has to be careful with Soulfire because he could accidentally kill himself with it. He also anticipates that, should he do so, his soul will simply be gone. No ghost, no afterlife, just a cessation of existence.
Thus, in your hypothetical situation, a wizard in a near-death experience who used his Death Curse would die. He would die despite anything healing his physical wounds. In Dresden Files, magic is portrayed as the source of life, and the Death Curse expends it all. You could fix the wounds, but it wouldn't transform the wizard's body into a living one.