Protect Inverter from Transformer's Inrush Current
Magnetizing current flows through the primary winding of a power transformer to energize the core.  This is independent of the loads connected to the secondary winding and is voltage dependent.

When approaching primary saturation voltage, from high input voltage;  


the primary current harmonics (5th and 3rd) begin to rise 
the inductance drops 10% at threshold of saturation 
thus reactive current rises 
and lowers the input impedance. 


When the AC voltage is switched off , there is likely to be a Remanence or magnetization charge in the core that remains for a long time. Thus if voltage is applied in the different phase crossing from the previous shutoff the magnetization can get out of sync and if the core does not have enough margin to saturation may cause a sharp drop in inductance until it self-balances.  ABB makes smart reclosures ($) to resolve this issue for large XFMR's.

If I understand correctly, instead of 10% excitation current you are getting 300% to 400% rated current (or perhaps just normal operating current) on startup. This is likely due to core saturation from Remanence. 

This tells me something about your nasty load but leaves me guessing about my assumptions and your measurements for margin to saturation, X, L/R and, harmonic content, load characteristics and other ratings.

A little more info on your loads and transformer tests will help suggest a better cure.

It this surge is purely motor-load surge then it is not "transformer surge" from Remanence, Core, Saturation, overvoltage etc.

Your options then are ;


Choose an HMCP circuit breaker with a higher instantaneous trip range.
Substitute a thermal magnetic circuit breaker with a higher instantaneous trip range.
reduce tap voltage