My suggestion is that the character take a stunt related to manipulating consequences, rather than stress. Also, it should be a limited ability, either requiring resources to use, or having a set maximum number of uses. A character that simply doesn't take stress will, as you say, break the combat system, and being able to ignore pain doesn't prevent your body from being *damaged*. As a starting point, consider the Last Leg stunt from the [Fate Core website][1]:

> **Last Leg:** You may spend fate points to keep standing. Any time you
> would suffer a consequence or be taken out by physical stress, you may
> spend a fate point to remain standing or otherwise defer a consequence
> for one more exchange, or until you take stress again, whichever comes
> first. Once the extra time you've bought is up, all the effects you
> have deferred come to bear at once. You may keep spending fate points
> each time the time limit expires until you run out.

Personally, I think this is a bit weak for a stunt - it requires a Fate point to power it, and only lasts for one exchange. DFRPG powers often give significantly greater benefit-per-refresh than this, so I would suggest making something that uses a similar mechanic, but is better - allows the character to defer the consequence longer (in keeping with your description). Perhaps something like this:

> **Ignore the Pain:** You may spend fate points to ignore injury... for a while. Once per scene, when you would take a physical consequence,
> you may spend a Fate point to defer that consequence until the next
> day (or the start of the next session, whichever comes first). If you
> are unable to take the consequence when it returns, it rolls up to the
> next higher consequence slot. If you cannot take it at all, you are
> taken out at that time.


  [1]: http://evilhat.wikidot.com/fate-core-stunts#toc84