Not sure I believe the manual on that, but I could be wrong. I think as long as you stick to a 2S lipo it shouldn't hurt the ESC, as it won't be over-volting it and the ESC decides how much current to draw from the battery.
Re. the C rating, this is just a rating of how much current the lipo can deliver at one time, so if your motor/ESC is getting as much current as it wants then a higher C rating won't make any difference. You can do the maths on how much current the motor can pull at max, but my bet would be that it's not going to need more than a 20C lipo to deliver that. A 4000maH lipo with a 20C rating can deliver 80A peak current, which is plenty for a simple brushed setup. If you can borrow some different C rated lipos and test back-to-back then you should be able to confirm this, but all I'm saying is don't go out and buy a lipo with a high C rating thinking it will give more power. The battery can be a bottleneck for power delivery, which is why the lipos improve things, but the motor/ESC is what defines the maximum power you can deliver to the wheels.