No, increasing C rating will not increase capacity (or minimize power consumption).
C rating is related to the output resistance (or internal resistance IR) of the battery pack.
A higher C rating correlates to the ability of the battery to deliver higher output current without significant internal temperature rise
(Power = Currrent ^2 * Resistance....Current is the output current and Resistance is the battery pack IR; higher internal power dissipation leads to higher internal temperatures, which will shorten the battery life)
Higher C rated packs will be heavier than lower C rated packs simply due to the differences in construction/materials needed to lower internal resistance.
You could try a larger battery (5400mAh packs) or try a ThrottleTech in the airplane.