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).
This battery pack contains a protection device to protect the battery from overcharging, over-discharging and over-current. You're load seems to trip the over-current protection build into the battery pack. I suggest you do not try to do something about it as it might result in big explosion. NEVER use this battery without it's protection circuit.
(the circuit contains some MOSFETS and things like that and they might not be rated for your demand.)
Finding the capacity of batteries at different discharge rates is best done with Peukert's law that you can find here:
http://en.wikipedia.org/wiki/Peukert%27s_law
The wiki page mentions lead acid batteries. But it can also be used for Li-Ion batteries. However, getting the right numbers from a battery manufacturer can be a problem.
A more practical explanation can be found here:
http://www.bdbatteries.com/peukert.php
Regards, Hendrik