The NEMA MG1 standard requires that a motor be able to deliver rated output with a voltage unbalance of 1%. I don't have the standard in front of me; but, my recollection is that the NEMA MG1 standard defines the derate for a running motor down to a maximum of 10% voltage unbalance (you can't run a motor at rated output with a 10% unbalance). I don't recall that the NEMA standard limits the maximum unbalance to 10%; but, as I recall the derate is very severe so the effect may be that the motor is not operable with a 10 % unbalance.
The NEMA standard is based upon unbalance in the 3 phase to phase voltages, not the motor current. According to the NEMA standard, I recall that each 1% unbalance in voltage can cause 6-10% unbalance in the currents, so a 6-10% unbalance in the running current would be within reason if everything else is 'normal'.
As you note, the standard applies to a running motor. Measure your phase to phase voltages and use that to assess whether you have an unbalance supply problem. If your supply voltage is balanced within 1% and you have a current unbalance in excess of 6-10%, then you may have a problem with the motor. If the voltage supply balance on the running motor exceeds 1%, then time to start looking for a supply problem.
If this motor is started across-the-line, the peak instantaneous starting currents can be in the order of 6 times the running current and the peak instantaneous values in the three phases can measure out significantly different between phases, especially if you are using something like a digital multimeter with AC clamp and an instantaneous capture feature. The consistent A phase unbalance might be due to timing in the closure of the three poles on the circuit breaker or the way that pole is closing.
Is your circuit breaker operated by something like 51 and 50 relays? Make sure that your peak calculated starting current (6 times or whatever the appropriate in-rush is) is not encroaching on the 51 relay's inverse time characteristic or exceeding the set point on the 50 relay.