Minimalistic RTC backup battery charging or a super-capacitor?
In all the times I have designed an embedded system with a need to have a power backup for an RTC it turned out that after all the analysis was completed a non-rechargable coin battery was the superior choice. 

I've used CR2032 coin cell for this many times in a socket. Vertical types are available so that the PCB footprint is minimized.

Other products I have used a smaller battery like a CR1225 that is accommodated in a very simple socket like shown below. Although I use the through hole version of the socket bracket to save even more PC space.



Picture source: https://www.adafruit.com/product/1141&lt;/sup>