The instructions on these two devices (A flashlight and handheld radio) say not to use rechargeable batteries. Will using rechargeable NiMH batteries damage these devices or just cause them to malfunction? Both items sell a version with a rechargeable battery pack. 
NiMH batteries have lower initial voltage and lower internal resistance.
Lower voltage won't allow the device to operate properly. Not likely since alkaline voltage decreases over the useful life. A well-designed device should handle a lower voltage.
Lower battery internal resistance will stress the device. Rare, but some devices rely on the internal resistance of alkaline batteries. I have a toy with a fan that falls in this category.
Lower battery internal resistance is a safety hazard. NiMH batteries are powerful, a short will cause 10-20 amps or more to flow, enough to melt the insulation off the wires inside the device, and cause burns, or worse. To save a little money, they didn't include safety protection for device shorts with NiMH batteries. Presumably, the optional battery pack has protection built in.
#3 is the most likely reason for the warning.
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.
The danger is that the SBT-12 could cause damage to the batteries, not necessarily that the SBT-12 itself would be damaged.
Both items sell a version with a rechargeable battery pack.
The battery pack probably has the addition safety components required to protect the batteries.
