I'm 100% positive though, that a lithium battery cannot be used with the stock rectifier regulator.
This is untrue. It is recommended, and a better choice to get a Lithium compatible Reg/Rec, but it is NOT required.
Shorai is a popular Lithium brand, some have seen issues with AntiGravity (not everyone), but there's many different brands. What you should calculate is the capacity you will require with your setup and charging system.