The problem with something like the F02CT is that while its output power appears to be ok for this, it is designed for a continuous low-current load rather than a very short, very high current. One solution is relatively large capacitors on the outputs.
16 cycles at 1 MHz is a 16 microsecond current burst. Using a linear approximation for capacitor discharge, that works out to a 3.4 V sag in capacitor voltage with a 4.7 uF cap. IOW, if the DC/DC converter charges a pair of 10 uF output capacitors up to +/-100 V, then the capacitor voltage will sag down to something above 95 V during a burst. In the datasheet there is no maximum output capacitance spec, which is good.
How constant does the peak voltage have to be during a burst? The solution might be your DC/DC converter followed by a linear regulator. The datasheet says the output is proportional to the input, so running it on 13 V instead of 12 V should increase the output voltage enough to cover the regulator headroom. The regulator doesn't have to be great, but it has to be fast.
Or something like that.
ak