err, not entirely true. (but mostly)
You are looking at things wrong. The POTENTIAL current output of the battery has no bearing on the esc it is used on. The motor is what draws that current and that it what you base your esc on. You just need to make sure the battery you use can supply the needed current and technically the bigger the better as long as it fits.
Only point I would like to make is that a good battery that has a lower internal resistance will have a higher voltage at the higher currents. It will be able to keep up w/ the motor. So when your throttle is maxed out, the better battery will be able to supply the demands of the motor. The ESC is like the pipeline between the two. The ESC needs to be able to deal w/ that current delivery or it might die.
When I did some of my battery testing, I put 2 parallel 15C 3S batteries together and my current went from 25A to like 36A at full throttle! The battery was able to deliver the current the motor wanted without dropping voltage and this exceeded the 30A continuous current rating of my ESC. I knew I'd be ok since I never flew full throttle on that scale heli even though I flew it w/ 2 batteries in parallel.
I'd say that your fine so long as you are not flying fast forward flight at full throttle for minutes at a time and you don't have a mechanical issue that will draw more current than necessary. Well if you do, then your existing battery could fry the ESC as well.
A stock Belt CP (at least mine) does not draw alot of current anyways (less than 20A I believe) so I think your good.