I can confirm similar but not as bad behavior with a 1.4V input, 150ohm load (resistor box), and a Rigol DP832 PSU input.
I really don't think it is instability. It is "by design" intentional behaviour. At 150 Ohms load, the regulator is going to sleep (to save current) and waking again. This is a common method converters use if they are attempting to have a high efficiency over a wide current range.
You can read about it in this Linear Technology datasheet (Burst mode).
http://cds.linear.com/docs/en/datasheet/35392fc.pdfOther converter manufacturers (like Seiko Instruments) use exactly the same method to get efficiency over a wide range. the difference between LT, Seiko and Batteroo is that Batteroo let the voltage drop much lower then the other manufacturers before turning the regulator ON. That might be a tradeoff between quiescent current and p-p burst mode waveform. The less often the regulator wakes when no load, the less quiescent current.
What happens with a low impedance battery is it allows high peak currents so the regulator ON time is much shorter then for a higher impedance battery.
Basically you cannot have a switching converter running continuously at 2.5MHz and switching current capability of several amps and have a quiescent current of less then 10uA. The way you achieve this is by turning the regulator off for most of the time at less then 10% load.
150 ohms is a current of 10mA. On the waveform, I am seeing about .1V drop for 80uS. Assuming the regulator is off for this period, that means the output caps of the AA battery Batteriser total 0.01 X 80uS / 0.1V = 8uF.
The no load graph shows a 0.15v drop over 0.155 second, so with a 10uF output capacitor, the actual load current for the "no load" test was 8uF x 0.15/0.155 = 7.7uA. Assuming that is not the scope probe current, it means most of this is due to the regulator chip in the Batteriser (the output voltage sensor circuit).