Question on current sudden increase

Is the motor at rated speed when the transfer occurs? If not, the situation resembles DOL starting, with increased current.

is motor acting as load?

switching surges

voltage stored in coil etc. my be the reason.

Is there power factor correction in the inverter?

If the transition is open, for example with a transfer switch, ANY small amount of deceleration in the motor while there is no power on it during the transition will result in a current surge when reconnected.

If the transition is too fast, the magnetic field in the motor may not have collapsed yet and the motor may still be regenerating. If that happens, when you re-connect, you will be out of phase with the supply and extreme current and torque surges may actually damage the motor.

A modern inverter gets feedback from motor's current vector and applies appropriate corrections depending on expected and actual load dependant slip, making it possible to apply less current for lighter loads, according to real conditions and programmed data. I assume you are referring to this current difference and not the expected surges when any disruption on normal power feed would cause, because of even momentary phase sequence loss and resulting de-synchronization. So generally some current variation is to be expected. S.M.

How long does the increase last for?

If your inverter puts out a square wave the effective voltage coming out of the inverter may be higher. If you are connecting to a lower voltage supply but providing constant power the load will draw more current. Also if your inverter is supplying a square wave and you are not using a true RMS meeter the reading will be off when looking at the inverter output.

I'd also be making sure you are using a true RMS meter, as indeed some inverters do produce square wave instead of the reasonably clean sine wave from the mains.

The current increase may be measurement error due to this

because there is a speed and / or frequency mismatch from before to after.

Some inverter control can be less 'stiff' if the applied voltage is lower than the mains supply or rating (aka energy saving mode for example) so the reconnection accelerates the motor to its load rated speed from the lower speed of the VFD. Maybe this could be your unexpected increase in current at what you may think is the same speed?