+1 for soft-starting the load if you can.
In principle, you can make a power electronics box to supplement the d. for load peaks. This would pretty much be a STATCOM device with a nice big energy storage capacitor. To get fast response, you would likely use a simple inductor output filter (not LCL) with a high switching frequency (probably > 20 kHz) and some cunning control design.
However, this is quite complicated to design and may interact with the inverter control systems, especially if using voltage-only control. For example, the STATCOM and inverter could fight about what the appropriate voltage level is, or they could oscillate against each other until something goes bang.
If you measured the load currents you could implement a current compensation scheme, which could help with control design.