I can't say anything about that particular controller (I don't own one), but I can shed some light on why they might be recommending against rechargeable batteries.
While the controller doesn't really care about the particular chemistry of the battery you put in it, it might care about the battery voltage.
Regular alkaline non-rechargeable batteries come in at ~1.6V fresh from the box and have a fairly linear discharge curve. NiMH rechargeable cells have a voltage of ~1.4V when fully charged and sit between 1.1V and 1.2V when discharging most of the time.
Some appliances work great with both types while others refuse to work at the 1.2V level, which means that NiMH batteries don't really suit them - you can't access most of the energy stored. That might be the case here.
There is a third option though - you can look for lithium based rechargeable AA batteries. They have a built-in buck converter and USB charging circuitry. You can get them from e-bay or amazon, though they are more expensive than NiMH batteries, so stick to those if they work for you.