My suggestion is that the character take a stunt related to manipulating consequences, rather than stress. Stress is ephemeral - it vanishes after the combat is over, and unless you're in the middle of a combat, taking stress is basically meaningless, which means that 'delaying' stress doesn't really work. Consequences, on the other hand, represent the lasting effects of injury. 'Being in pain' is one of the standard lasting effects of injury, so being able to defer consequences fits reasonably well with the idea of deferred pain.
Also, it should be a limited ability, either requiring resources (fate points) to use, or having a set maximum number of uses (or both). A character that simply doesn't take stress or consequences 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:
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.
In addition to keeping the consequence slot open, this has the benefit of denying the enemy who would have inflicted a consequence their free invocation thereof, at least for the duration of the fight.
My assessment is that, balance-wise, this is pretty strong for a 1-refresh stunt, but weaker than a 2-refresh power (like Inhuman Recovery). Depending on how much refresh the character wants to spend on their 'deferred pain' ability, you could ramp up this power in a variety of ways: remove the fate point cost, allow it to be used more times, defer the consequences longer, etc. It could also be combined with one of the Recovery powers, which would make it much more likely that the character can clear out their other consequences by the time the 'deferred' one(s) arrive.