Transformer overdraw behaviour
As you mentioned, there are two main problems with overloading a transformer:
1 - Overheating
2 - Voltage drop
Apparently you already have an estimate on both issues, but a detail about the transformer is that when you require 160% of load, the losses in the windings will increase to 256% of the rated losses, that is, the transformer can heat up a lot and the life time of the insulation can be drastically reduced. Of course, the time that it is under low load can compensate for this excessive loss of life during the overload.
About core saturation. In general, there are no problems. The flux densities in the core columns depends directly on the voltages, in which case I understand that you will apply the rated voltage. Due to the overload, the flux density can increase in some points of the core (since the stray field is added to the magnetization field), but the voltage drop, in this case, reduces the magnetization field and generally the thing remains far from saturation .