NiCd starts at 1.2V. But nobody (should be) use(ing) those any-more. My experience is that most good NiMH cells start at 1.4-1.5V and maintain 1.25V during most of their in-use-time.
It shouldn't be too hard to switch on the boost circuit this Batteriser supposedly has, only when a reasonable current is drawn by the device. A current sensing circuit should not add that many components. That should prevent the Batteriser from drawing power to keep the boost circuit running when the device is switched off...
-Unfortunately I'm unable to show you the ASCII art I had made... So here is a description of it:
Main circuit, in series: +bat --[>|-- [R2] - device - -bat (ground).
Booster circuit, 4 leads: on both sides of the diode, ground and Current detection sense input.
Current detection circuit, 3 leads: on both sides of R2 and Current detection sense output.
Something like this (just a rough idea. I'm a radio amateur, not an electronics engineer): R2 is very small and in series with the device (and the battery). Current is sensed over R2. A diode (pref. a low drop one) maintains most of the original battery voltage over device as long as little current is drawn. When boost circuit kicks in, voltage is boosted, the diode prevents the surplus voltage from running back to the input of the boost circuit. Drawback, R2 will dissipate some energy and its resistance depends on how sensitive you can make your current detection circuit. Also, the current detection circuit itself may use some energy but if devices with a 'soft' standby can be battery powered and drain the battery in a matter of multiple years instead of days (I'm looking at calculators and those small bike LED-lights), so can this Batteriser.
There is only one sort of device I can think of which may not be able to deal with this. I know some devices that have a stand-by current in the micro-ampere range that still do need a rather high supply voltage, else they reset, restart, draw lots of current while restarting and then go back into stand-by. If the booster circuit shuts down while they are in standby, it would mean and endless cycle of restarts, draining the battery even faster. Of course those devices already would drain the battery faster in the original situation. When the battery voltage would become low the first time, the device resets. Then because of the internal resistance of the battery, the voltage would drop even more by the current drawn by the start-up sequence.... resulting in a reset