Batteries operate by chemical reactions. One property of the chemistry determines how much charge the battery holds leading to your amp-hour spec. A 2nd property determines how fast the chemicals can usefully react limiting the maximum surge current. A battery is not a capacitor. You cannot draw current faster than the chemistry without damaging the battery.
In general people are using hi surge LiPo batteries. LiPo's sold for RC planes and helicopters often quote the max safe surge current.
If size and weight are unimportant, you cannot beat the lead/acid battery.