OP is "Original Poster".
I think your best bet is to modulate the heater to start over time.
There is a simple PWM controller at Jameco. e.g
https://www.jameco.com/webapp/wcs/stores/servlet/Product_10001_10001_120539_-1 You would just have to make a MOSFET driver able to carry 200 Amps or so. Inherently, the PWM driver contains a soft-start circuit, but you may need to ramp the setpoint.
Would you happen to know the material that the heating element is made out of? Can you find it out? Wire AWG, length and material?
Are you capable of measuring the cold resistance of the element? You would need to be able to measure the current through the device and the voltage across it. As long as you have the resolution in voltage, you may be able to determine it.
So, I see two solutions:
1) A high power resistor. I once bought a 1000 W 0.5 ohm adjustabe resistor, but they are expensive. The downside is the contactor will draw current too.
2) PWM and create a slow turn-on for the heating element
3) An inductor in series with the heating element. An inductor won't allow the current to change instantaneously.
4) Resize your inverter
This really looks like the case that the cold resistance of the element is too low and exceeds the VA capacity of your inverter. Only #2 seems practical.