The chances of any adverse safety event are very small. The only concern is voltage compatibility. If they designed the device with a relatively high cut off voltage, then the NiMH cells may give short battery life. I have some little portable radios that are that way.

But I have also designed half a dozen AA powered toys and we ALWAYS designed the low-voltage cutoff so that the toy would work well with NiMH batteries. So you may have to try it to see if there is any problem. Since they sell a rechargeable pack, I guess it likely WILL work with NiMH.

I can't tell you that there is no risk of damage or injury etc. But I can tell you that I would not hesitate to try it with NiMH cells. Also, I think it is bad form for any company to insist that you only use disposable cells nowadays. So complaining to customer service may be in order.