As for damage while in the mechanic's possession: a flat battery is NOT damage. Heck, if that was the case then whenever you needed a new battery you could simply drive into any shop for an oil change and not pick up the car for a couple of days. Then when you do pick it up, OH you damaged my battery! You pay for a new one.
If the battery was NFG when he went in, it will be NFG when he comes out unless he orders a new one.
The ONLY way the OP shouldn't have to pay for the repair is if the mechanic didn't actually replace the timing chain. Other than that? There's no reason not to.
As I stated, just because during the investigation of this "noise" the repair recommended isn't the culprit, doesn't mean the mechanic should foot the bill for the new timing chain.
Put it another way: you own a house, find a water stain on your ceiling, call over a contractor to find it and fix it. He finds a suspect pipe, repairs it. doesn't see any other source. A week later you find another water stain, he comes back, finds another suspect pipe that was hidden behind a stud and under insulation, fixes it. A week later you have another water stain, he comes back, finds a leaky boot over a vent stack, fixes it.
Do you only pay for the one fix or all of them? Cuz if you have more than one leaky pipe then even if only one of them is making the stain, they all should be fixed. Just like the timing chain: if it is worn and making noise, it should be fixed even if it isn't the particular problem.
BTW: I come from a family of mechanics and my brother worked at a dealership. He worked 40 hrs on a brand new car with less than 1,000 k on it for a rattle. They tried everything. Then eventually found a nut under one of the rocker panels. (back when there were rocker panels). Ford covered his entire week under warranty.