Hello!
Just for info, I used the following implementation (see picture).
I didn't use a battery, but a capacitor.
Its capacity is 0.2F, and it could run a clock for more than a week.
Advantages: no battery charge circuit needed. A resistor and a diode is enough.
Exteremly small, less than 2mm high. 6mm in diameter.
NB: It was made by Panasonic, but they stopped or will stop the production.
There are alternative at Digikey. Look for "supercap".
NB: in this case, diode wasn't even needed. I was using a RTC1390 from Maxim.
This chip manages the charge by itself.