Have you tested the battery to ensure it is in good health?
I've used rechargeable AA and AA in mice for many years without significant problems, BUT the ones that use 2 x AA tend to be a bit heavier than I like. I will even take the weight slug out of corded mice to make them lighter. However certain mice, some logitechs come to mind, may use two cells but they are electrically in parallel so you can run the mouse with only one battery installed and just have half the battery life till drained.
A mouse with a built in battery tends to be lighter weight, tends to cost more, and unless you're into electronics including soldering and sourcing parts, once the built in battery's lifespan is over, you're going to have to replace the whole mouse if soldering in a new battery is outside your ability.
Since you already have the mouse, I'd try a different battery, something major brand and Low Self Discharge will work best in a mouse.
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