Which battery type for high power and fast charging?
Selected versions of -


Lithium Polymer. "LiPo"
LiFePo4 / Lithium Ferro Phosphate  
LiIon only if 4 strings used at 24V (or 2 strings at 48V).
A 40 Ah lead acid battery (large car, small truck size) would be up to this with an inverter if you didn't want too too many life cycles.
A 12V 100 Ah deep cycle battery would be happy. 


Specific specifications need to be checked but it is common to find LiPo cells with 10C charge rates.

You don't say what voltage you want.
 I'll assume 24V, although that's probably low.
 200W for 10 minutes = 200 x 10/60 Watt.hour ~= 33 Watt hour.
Say 50 Wh of storage to be safe.
At 24V you'd need about 2Ah of capacity which is what you'd get from an 18650 LiIon cell (but the discharge rate is too high) , or  a LiPo of equivalent capacity.
 So, using a ~= 3V endpoint your need could be met by say 24/3 = 8 x LiPo 2Ah+ cells.

Discharge rate = I = P/V = 200 Watt/24V ~= 8A = 4C for a 2AH cell. 

You don't want to be deep discharging your cells in 10 minutes as lifetime will be impacted (and audiences may suddenly be un-sounded so 2 or 3 strings in parallel would be wise, or larger capacity cells.