I guess my game was not going to have it, with me not updating my signature. In less than 150 eggs, it spat my target shiny, Cutiefly, at me. And it couldn't have given me a better one. Female, Shield Dust, Timid, Love Ball, egg moves Baton Pass/Moonblast, 31/x/31/31/31/31.
Onwards to Wimpod!
EDIT: What on earth?! Third egg, Wimpod shone! I quite literally yelled "WHAT?!" Male, Lure Ball, Adamant, egg moves Wide Guard/Aqua Jet/Spikes, 31s in HP/Def/SpDef/Speed, "very good" in both attack stats.
Next target is either going to be Meowth or Cubone.