> Let's say there's a 2Ah battery with a C rating of 2C.

The above statement is incorrect.

The correct statement would be 'Let's say there's a C2 rated battery having a capacity of 2 Ah'. It would refer to a 2 Ah battery capable of delivering 1 A maximum for 2 hours without getting damaged.

Consider another example of a C20 rated battery having a capacity of 100 Ah. It would refer to a 100 Ah battery capable of delivering 5 A maximum for 20 hours without getting damaged.

It should be borne in mind that the rated capacity would not be fully available at higher discharge rates.

> How do can I increase the current discharge for this particular battery?

The discharge current of a particular battery would be dependant on its voltage and the resistance of the load.