I don't know the ins-n-outs of the Game Changer power requirements but...
Even when fully charged, rechargeable batteries give a lower voltage than standard batteries. Standard AA cells are usually 1.5V when fresh and then gradually decrease in voltage as you use them. NiMH rechargeables don't have the same gradual decrease in voltage of normal batteries, they spend most of their time at only 1.2 V, before they drop in voltage very quickly at the end of the charge. Disadvantages are the lower initial voltage plus less warning before they need changing.
For analog electronics, it's not a big deal - usually the sound quality/output gradually decreases until you notice your tone sucks and you change the batteries. For anything with digital components there may be a minimum voltage required to maintain proper function. Below that voltage level, normal function is impossible.
The question is this - is 1.2 V enough? Even more importantly, the Game changer uses 3 x AA, so, will it function OK at (3 x 1.2 V) = 3.6 V, instead of the 4.5 V it would like to see?