In the meantime, I'm going to great lengths to get the best Pokémon for my team... in their appropriate Pokeballs. Thus far each of my Pokémon have perfect IV's in their relevant stats, as well as the correct natures and right moves. This is what I've got so far:
Timid Espeon in a Heal Ball, with Stored Power (Done completely from scratch)
Impish Tyranitar in a Moon Ball, with the egg/tutor moves Dragon Dance & Iron Defense (Bred from a random Larvitar I got in a trade, along with an Axew I had -- all from scratch)
Sassy Azumarill in a Dive Ball, with the egg move Aqua Jet (Done completely from scratch)
Hasty Diancie in a Cherish Ball (compliments of a trade) -- Plans on replacing this one with a better-natured one are coming smoothly.
This is what I've still got left to obtain:
Impish Shellder in a Heavy Ball, with the egg move Rock Blast (I'm having to play through my Heart Gold in order to obtain the Super Rod to catch a female in this particular ball. This is gonna take some time.
)
Timid Larvesta in a Repeat Ball, with the egg move Morning Sun
Timid Diancie -- Plans on obtaining one soon
EDIT: While I'm at it, I'm probably going to end up catching a female Magikarp in a Heavy Ball as well... Because having a Heavy Ball Gyarados running amok sounds like a really cool idea to me!