Pumping a few amperes for 100 µsec
That's an average power of 

Power = 5 A \$\times\$ 10.5 V \$\times\$ 100 \$\mu\$s / 10 ms = 0.525 W.

Average power is easy for almost any battery. You just need a store to accomodate the pulse. 

A capacitor that will "droop" say 0.5V in 100 \$\mu\$s needs to be

C = I \$\times\$ t / V = 5 A \$\times\$ 100 \$\mu\$s / 0.5 V= 1000 \$\mu\$F.  

A supercap would do well here if voltage rating is OK.

E&amp;OE