Can I convert L-ion battery backup to supercapacitor?
Just substituting a supercap for a LIon is not going to be good.

A LIon dies if the voltage exceeds 4.2v, the charger will limit to handle this. A supercap dies if the voltage exceeds 2.7v. Oops!

Two series supercaps, with DC balancing resistors to protect each against overvoltage, might work OK, but with the charger stopping at a max voltage of 4.2v versus their capability of 5.4v, you won't use their full potential. 

Cost for cost, a LIon is likely to have a much higher usable capacity than a supercap. As long as it stores enough energy for an orderly sync and shutdown, that difference may be OK for you. If the controller thinks it has a LIon connected, then it may over-estimate the energy it has remaining at its disposal, and fail to start the orderly shutdown soon enough.