Pure Speculation Follows
By Iron Man 2, Tony Stark is using the Arc Reactor not just to keep the shrapnel out of his heart, but to power Iron Man and, by his own lights, to protect the world:
Tony Stark: The point is you're welcome, I guess.
Senator Stern: For what?
Tony Stark: Because I'm your nuclear deterrent. It's working. We're
safe. America is secure. You want my property? You can't have it. But
I did you a big favor.
[stands and turns to face the Senate]
Tony Stark: I've successfully privatized world peace. What more do you
want? For now! I tried to play ball with these ass-clowns.
For Tony to trade in the Arc Reactor for a battery would keep him alive, but would destroy both the Pax Ferrei and (more importantly) Stark's reputation for brilliance and blatant disregard for consequences. The mythos of "Iron Man" would be damaged.
Is Tony Stark so prideful that he would rather die than look like his brilliance had failed? Given his behavior throughout the movies.... yes.