The C rate is limited by the battery internal resistance, which leads to heating during discharge. If you add cooling to the battery it can sustain a higher discharge rate, but you should consult the manufacturer.
More about C rate here: https://batteryuniversity.com/article/bu-402-what-is-c-rate
That said, exceeding the recommended discharge rate for a Li-ion can damage the battery and possibly cause an explosion. Most Li-ion batteries sold to consumers have protection built in to prevent this. Nevertheless, there’s plenty of YouTube videos of Li-ion batteries gone wild (especially things like vape pens - go figure.)
If your load uses a lower voltage than the battery set, you can use a step-down regulator to increase the current. This lowers the discharge rate, so you could possibly get more run time, depending on the conversion efficiency.