How do I find the maximum "discharge C rating" of a battery?
You're not really looking for the C rating (maximum discharge current in multiples of nominal capacity), you're looking for the adjusted capacity at your nominated discharge current.

A 1.8Ah lithium battery can theoretically give 1.8A for 1hour, or 3A for 1.8/3h = 36 minutes.   HOWEVER the capacity for a battery is traditionally quoted for a 20 hour discharge.   That is, a capacity rating of 1.8Ah means the battery delivered 90mA for 20 hours in testing. 

The relationship between continuous current and time-to-full-discharge is NOT linear.    You have discovered that when discharged at 3A, your 1.8Ah battery is delivering much less capacity (only about a quarter!) than a linear interpolation of the amp-hour rating would suggest.    This is not unusual.

The better batteries will give rated capacity at several discharge time samples, or even a graph of current vs capacity.   A battery intended for remote control uses will probably give better performance, as quick discharge is the intended application of these batteries.

(I use a 1500mAh 3-cell (11.1v) 25C lithium pack to power a 3A LED bike light, and I get around an hour, which is close to rated capacity given a reasonably efficient buck regulator).