First what OEM converter did you buy.
Second what is the charge/float voltage for your LI battery
As I recall (And I stress that As I recall part for I may recall incorrectly) LI's float and charge a couple points higher than Lead Acid.
Google says A fully charged 12V LiFePO4 battery will have a charging voltage of around 14.6 volts and a resting voltage of around 13.6 volts.
For Lead acid Google says: The typical charging voltage is between 2.15 volts per cell (12.9 volts for a 12V 6 cell battery) and 2.35 volts per cell (14.1 volts for a 12V 6 cell battery)
Which is not what I recall from earlier research. (I recall closer to LiFePO4 specs)
In either case I do not think the LI will be damaged by the OEM converter.