Quick answer.....no.
The "C" rating determines how much capacity (milliamp rating,2200.2500.etc..) the pack can provide at one time without damage to the pack. They call that " burst" rate. A pack with a higher "C" rate will provide the motor with the 2200ma quicker than a pack with a low "C" rate.
The motor and prop size or in your case the fan size determine how many amps will be drawn from the pack so you need to install an ESC capable of supporting that amperage rating and that has been determined when it was built to be 45amps.
So in your set-up the motor/fan can only draw a certain amperage at full throttle. Installing a pack with a higher "C" rating can not make the motor draw any more amps. What it will do is provide the motor with it's required amperage for a longer period of time at full throttle than a pack with a lower "C"rate. Be aware that the higher you go in "C" rate the larger and heavier the packs get ; the same as if you increased the capacity from 2200 to 3000mah. Also increasing the capacity won't affect the amp draw either.
To put it another way, if you are watering your lawn with a bucket of water and a hose. The bucket represents the capacity and the diameter of the hose represents the "C"rating. Larger capacity means more water in the bucket and a larger "C" rating means a bigger hose so you get more out, faster.