insert in series with your load a 47 Ohm resistor that is shorted after the inrush by a delayed relay. A better alternative would be some fabricated soft start circuit, such as a NTC resistor just like in the PC power supplies that also have an inrush when connected to the mains voltage. A NTC from an old PC power supply at least should be tried. Just put it in series with your load.
If you need that 100 ms period + 100% extra power, then the extra must be stored to somewhere before your load is connected. In theory that can be a LC resonator, just in perfect tune for 400 Hz. But that requires so huge and low loss coil that it would be impractical. A better alternative would be a a motor with a substantial flywheel. The motor should be a sychronous one to avoid the phase difference. Its uprunning would be tricky.
What about an active electronic circuit that emulates the motor? Forget it! It woud be an extra power supply.