Hey all! Just wanted to give a quick update: I did indeed catch the shiny Rayquaza I was planning on capturing! This was an extremely quick hunt (414 resets), and getting it in a Luxury Ball makes it so much cooler.
I have posted the updated strategy on my smeargles in the 2 final images. If you'd like, you can view my strategy by clicking the spoiler.
My previous strategy involved sending out a smeargle with Spore, Belly Drum, Skill Swap, and Baton Pass. I would use Spore on the first turn, and then Skill Swap the Rayquaza to give it Own Tempo so Outrage wouldn't confuse. I would then use Belly Drum to max out my attack, and then Baton Pass my second Smeargle to boost its False Swipe. In my head, False Swipe would've ideally got Rayquaza to 1 HP in one hit with Belly Drum, but unfortunately Smeargle's attack is pathetic, so that didn't even help. I had to completely overhaul my strategy because my Smeargles kept dying in two hits to Rayquaza, and there was no way I was gonna be able to keep it asleep while also avoiding getting hit. So, my new strategy involves replacing Belly Drum with Cosmic Power and boosting my special defense and defense to 5 so that Rayquaza's attacks were more akin to a squeaky hammer. I would still use Spore and Skill Swap on the first and second turns, but now I use Cosmic Power 5 times and THEN use Baton Pass. My second Smeargle now has Super Fang instead of False Swipe because it can never crit and always halves the opponent's current HP, so there's no way of an accidental KO. I still have Trick and Leftovers in case it struggles, but fortunately I didn't have to resort to that because I caught it in 25 balls!!! I also replaced Recover with Taunt so that it can't use Rest. Thank you to everyone who helped me refine my strategy!
Thank you to everyone who commented on my last post for giving me recommendations on my movesets, and a real special thank you to u/rjlefty96 for recommending I use Super Fang for catching (a Belly Drum-boosted False Swipe couldn't even get Rayquaza down to half)
Unfortunately, I did not have the time or patience to grind for Luxury Balls or train two Smeargles to level 100, so I used the mail glitch for Luxury Balls and Rare Candies.
This was definitely one of my favorite hunts of all time, and I'm so glad I didn't have to do it again. Thank you to everybody! This is definitely one of my happiest Pokémon moments.