The first problem you encounter if you use an external 220V -> 110V converter to power the original Roomba/Scooba power supply is the output voltage quality (the 50/60 Hz rating is not critical).
If your converter outputs 110V RMS, this doesn’t mean that the peak voltage will be less than the maximum allowed (155 Volts) of the original power supply.
For example, with a Remington 220V -> 110V electronic converter, which outputs 110Volts RMS, the peak output voltage is more than 200V (without any load). In this case the converter would certainly blow the Roomba/Scooba power supply (please note that the peak voltage cannot be measured with a standard digital multimeter, but only with an oscilloscope).
The other problem is that if you fed the original Roomba fast charger with a voltage below 80V AC you can have an output voltage over the rated 22V DC.
For exapmle feeding the fast charger with 60Volts AC, it’s output would be at 40V DC and any connected Roomba/Scooba will be destroyed.
According to the TOP245 (TOP246 for Scooba) datasheet, the 1Mohm and 1.3Mohm resistors R1 and R2 (on the power supply board) at the Line Sense pin of the TOP245/TOP246 should shut it down if the input voltage falls below 100 VDC at the filter capacitor, but this does not happen. It seems there is a bug at the design of the power supply and at undervoltage conditions, the TOP245 can behave in a wrong way.
This condition (60Volts AC mains) would never happen in real life, but with an external 220V -> 110V converter it could be possible.