By RAW, only leveling.
Base Attack Bonus is a class feature, and it changes when you gain a level in some class.
Though there are prestige classes that advance spellcasting AND base attack bonus simultaneously, for example Eldritch Knight. Take a look around, there should be more of them, at least in 3.5 there were.
You also might want to adapt Battle Sorcerer class variant, which has a cleric BAB progression.
From the houseruling point of view, consider this: most classes and prestige classes are already kind of balanced in terms what abilities do they offer. Sure, there are less powerful and more powerful abilities, but generally speaking they are already fine, at least in BAB department. Surprisingly enough, prerequisites are also often well-written.
Now, we must also consider that the Arcane Archer was designed as a master of ranged combat (hence the high number of feats and BAB requirement), who dabbles in arcane archery. Sure, one might houserule the BAB requirements away, but that ceases to reflect the original design, so I would advise agains that.
To sum up, a balanced solution would be either
- Multiclass. Preferrably to a prestige with full spellcasting AND base attack bonus progression.
- Adapt sorcerer variant from 3.5.
- Roll with it.