Extra capacity batteries won't make any difference other than to allow use of the mouse for much longer periods before needed recharging or replacing.
Have you checked out the mouse and battery charger contacts are clean and that the USB lead and adapter wires are connected correctly?
I've have a similar Logitech mouse and charger and it will work for some considerable time without me having to put it on the charger (I regularly forget to replace it on the charger due to habit; my previous mouse didn't come with a charger and you had to charge/replace the batteries about every six to eight weeks).
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Charging cordless mouse 