We do not add any extra lithium source other than counter/reference electrode and electrolyte (LiPF6). But in case of aluminium ion battery, we do add AlCl3 to the electrolyte additionally.
The cathode/anodes you use for Li-ion might not work for Al-ion batteries. Li+ is a much larger ion than Al3+, however due to the 3+ valence of Al ion it has much greater coulombic interactions which will greatly slow down diffusion.
Also, are you using aqueous or non-aqueous electrolyte? The hydration of ions can help reduce the coulombic interactions, making Mg2+, Zn2+, or Al3+ ion intercalation viable; however, some cathode materials are unstable in aqueous environment. Additionally, the desolvation energy to completely desolvate an ion in organic solvent will probably be greater than partial desolvation of water during intercalation.
To my knowledge, the ions that constitute the electrolyte of a battery must be of the same nature as that of positive and negative poles. This can provide more efficiency to the battery. As you know, batteries are based on oxidation/reduction reactions. The latter strongly depends on the electrochemical potentials of couples that participate in the total reaction of the battery. Consequently, the choice of ions (electrolyte) and electrode material is a key parameter that controls the functionality of the battery.