This thread is for fun, but it does concern the type of strict vetting of the rules that I find under Rules Questions.
Arcane archer has some pretty tough requirements. The two that are most important here are:
Ability to cast 1st-level arcane spells.
So for a character to enter Arcane Archer and still gain spells from the class they have to have at least on level in sorcerer, Wizard, Bard or Witch. None of these have the fast progression for BAB.
So the earliest you can enter Arcane Archer is starting at 8th level(1 level arcane, 7 levels fast progression for BAB)
It would be really cool if you could get both fast progression for BAB and Arcane Spell casting.
In one of the AP there is a creature that qualified for Mystic Theurge using her race. Here race specifically casts as a class equal to her HD.
Her race is Nymph.
A nymph casts spells as a 7th-level druid, but cannot swap out prepared spells to cast summon spells.
You could actually enter Arcane Archer one level(I realize that HD and level are not exactly the same thing.) earlier if you were a Fairy Dragon!
Dragons have a base attack bonus equal to their total Hit Dice, and Fairy Dragons cast spells as 3rd-level sorcerers. Three levels to gain the feats and then Arcane Archer.