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I was planning on doing a Sword playthrough using the strongest Pokemon possible using these oddly specific requirements. Just curious to see, as Gyrados and Togekiss surprised me at how high their base stat total.

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Strongest in what criteria? Do you mean effectiveness in game's single player campaign.
I think it's Inteleon, Arcanine, or Excadrill, mostly because I've seen speedrunners use them.
I simplified the title. There are no megas in Sw/Sh and pseudo-legendaries have 600 BST by definition.

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Top 10 highest (No legendaries, Pseudo-legendaries, Mythicals, Ultra Beasts or Mega Evolutions) BST Pokémon (National Dex)

1- Slaking (670)
2- Wishiwashi (School Form) (620)
3- Archeops (567)
4- Arcanine (Both Forms) (555)
5- Florges (All Forms) (552)
6- Ursaluna / Volcarona (550)
7- Togekiss (545)
8- Haxorus /Darmanitan (Zen Mode(Unovan and Galarian)) /Magmortar /Electivire/Milotic/Snorlax/Blissey/Kingdra/Gyarados (540)
9- Lapras/Crobat/Swampert/Magnezone/Rhyperior/Tangrowth/Porygon-Z/Vanilluxe/Noivern/Duraludon (535)
10- Charizard/Typhlosion(Both Forms)/Infernape/Delphox (534)

Top 10 highest (No legendaries, Pseudo-legendaries, Mythicals, Ultra Beasts or Mega Evolutions) BST Pokémon (SwSh)

1- Wishiwashi (School Form) (620)
2- Archeops (567)
3- Arcanine (555)
4- Volcarona (550)
5- Togekiss (545)
6- Haxorus /Darmanitan (Zen Mode (Unovan and Galarian)) /Magmortar /Electivire/Milotic/Snorlax/Blissey/Kingdra/Gyarados (540)
7- Lapras/Crobat/Swampert/Magnezone/Rhyperior/Tangrowth/Porygon-Z/Vanilluxe/Noivern/Duraludon (535)
8- Charizard (534)
9- Blastoise/Exeggcutor (Both Forms) /Sceptile/Blaziken/Aggron/Walrein/Mamoswine/Decidueye/Incineroar/Primarina/Golisopod/Rillaboom/CInderace/Inteleon (530)
10- Venusaur/Cloyster/ All Eeveelutions /Lucario/Hippowdon/Probopass/Dusknoir/Centiscorch (525)

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Wow, this is perfect. Thanks!
np, glad I was able to help :)