Ah, I think this is related to the issue, which is vaguely similar to the output linear operation but it's to do with the input:
"8.2.2.1 Overload Recovery With Negative VSENSE
The INA293-Q1 is a unidirectional current sense amplifier that is meant to operate with a positive differential
input voltage (VSENSE). If negative VSENSE is applied, the device is placed in an overload condition and requires
time to recover once VSENSE returns positive. The required overload recovery time increases with more negative
VSENSE."
Really wish TI gave actual numbers here.
I think any input voltage of 0V + Vos (the 80uV or so) can be considered negative, depending on that opamp's input offset. I've biased the +ve input to 180 uV and now the issue is gone. I think in next rev I'll bias it to the 20mV/gain to be 100% sure, this will require extra series resistors between shunt and INA inputs..