I own one of these litte cute PC many years ago. If I am not wrong, you are trying to use “normal” AA size Ni-Mh rechargable batteries instead of the battery pack.
You are right about the little plastic stub on the battery pack which tells the PC that you have inserted a rechargable battery pack instead of normal AA size alkaline batteries.
What I did last time was to find a small piece of plastic (cut it out from various sources that you can find) which would fit in between the empty space between the AA size Ni-Mh battery and the detector on the PC's battery compartment. Once this piece of extra plastic pushed in the detector, your PC will charge the AA Ni-Mh batteries as though it is the original battery pack.
It does work. Give it a try.