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Hopefully this isn't too late for you, OP. I've beaten Stadiums 1 and 2 100%. Here's some tips:
Dragon Rage is legal in Stadium 1's Pika Cup, so this should already give a good idea on what to use. Arcanine, Gyarados, Lapras, Aerodactyl, and Dragonair are all legal mons that can use Dragon Rage. Arcanine and Aerodactyl will require trading from Gold, while Dragonair will require glitching since you don't have Yellow. Throw an Alakazam on there since you can never have too much Psychics in Gen 1. For Petit Cup, Abra, Gastly, and Farfetch'd are all solid mons. Abra's the special sweeper, Farfetch'd the physical sweeper, and Gastly's normal immunity really comes in handy. For Poke Cup, a trio of Alakazam/Tauros/Zapdos should get you through it, but a Swift Persian wouldn't hurt as a fourth wheel in case of evasion faggotry. For Prime Cup and Gym Leader Castle, Amnesia/Recover/Substitute/Psychic Mewtwo should single-handedly get you through them, but if you want back up in case something terrible happens then a Physical Mew and some other powerful mon like Snorlax should suffice. For all modes, always have a lead with Substitute since the NPCs will almost always attempt to status you turn 1. Don't feel pressured to bother raising up six mons, since I generally ended up having one or two mons who almost never saw use, since Gen 1 has a really centralized meta. Be sure to exploit the item dupe and box glitches to build up your teams.
As for Stadium 2's Little Cup, Abra, Onix, and Scyther are kings. Curselax + Alakazam w/all elemental punches will get you through Poke Cup. For Prime Cup and Gym Leader Castle, those two should still work, but having a Curse Lugia for back up wouldn't hurt. Alternatively, with Curse Lugia around, you can replace Curselax with Belly Drum Lax and score even quicker, albeit riskier victories.