Should I use li-ion CR1220 cell or a different chemical type?
A CR1220 is a lithium/manganese dioxide cell at 3V. It's a non-rechargeable coin cell battery with about 40mAh capacity.

A lithium ion cell is a 3.7V rechargeable cell with a massive range of capacities. 

Can your RTC operate on up to 3.85V without issue? Do you require increased capacity or rechargeability?

I would just go with the CR1220. You'll get a higher energy density from the Li/MnO\$_2\$ and the proper voltage.