basically, as stated in the title, I am looking for a battery around the size of a CR1632, with a capacity of around 60+ mA. the higher capacity the better, and rechargeable is a must, and the same with a round shape.
I don’t think that “high capacity” and “small” go together physically (yet). But at the current rate of progress you may not have to wait too many years for this to become a reality.
CR1632 is a form factor, not a battery type. There are rechargeables available with that form factor, such as
That battery is quoted at 40mAh. An example of a rechargeable coin battery with greater capacity (200mAh) is :
RJD2450 CDE / Illinois Capacitor | Mouser Australia