16 Cycles at 1MHz means 16 x 0.5uS current pulses from each rail.
So 2A for 8uS each rail total, worst case.
The basic rule for capacitor discharge (or charging) is 1 Farad will change by 1V per second per 1A current.
So scaling capacitance and time, a 1uF cap will charge/discharge by 1V per microsecond per amp.
A 22uF cap on each rail should droop less than 1V during the pulse sequence.
I'd probably go with 47uF or 100uF as a convenient value - just make sure the ripple current rating is high enough, and preferably use 250V caps. (Capacitor lifetimes are very short if run near any of their max ratings).
2A for 8uS discharge would hypothetically need 50mA for 40 times as long, 320uS, to recharge.
Due to the very small voltage...