Living Dead is tough to decompose. It basically consists of one "power" ("Death is a Nuisance"), one "catch" ("Corpse Body") and one mixed power/catch ("Dude! You're Dead!"). Together, they have a net cost of [-1]. But you'd like to take just the beneficial part, which means you need to figure out how much that part costs.
Take a look at "Dude! You're Dead!" You get a +1 to Intimidation, but a -1 to all other social skills (Contacts, Deceit, Empathy, Performance, Presence, and Rapport). That's the equivalent of at least five (six negative minus one positive) 'negative' stunts, if you treat a "+1 to all uses of a skill" as a reasonable stunt, which is questionable.
"Corpse Body" is also a significant flaw. Someone kicks your shin and you now have a "Sore Shin" that will last until you find a way to fix it, and short of some sort of magical reconstruction, its not clear how to accomplish that.
Based on the flaws combined into the power, I'd say that "Death is a Nuisance" is much more than a minor ability.
That said, there's a rather cheap way to kinda-sorta get this sort of ability, if your GM is ok with it. First, take your choice of Wizard's Constitution [-0] or Inhuman Recovery [-2]. Then give yourself an aspect along the lines of "I'm not dead yet!" or "Just a flesh wound!" or, better yet, High Concept that refers to your repeated failure to stay dead. Then, any time you are taken out by something that should kill you (but could conceivably fail to), you can now invoke your aspect to narrate your way out of dying. This would require that (a) your GM is ok with that use of an aspect, and (b) that you can come up with a reasonable way to survive. That means it works well for the times when you are peppered with a hail of gunfire, throwing you over the side of the bridge into the river (luckily the bullets missed vital organs for the most part, and you managed to get caught on a branch further downstream). It doesn't work so well when the local Warden lops off your head 'on screen'. Note that Fate points are less of an issue, since you'd generally be gaining a couple just by being taken out.
This is still very powerful, which is why I mention several times that you'd need to get GM buy-in.