Solution 1:

Oh, figured it out. To actually get to a menu and be able to do anything useful (like download Poké Transporter), you have to launch Pokémon Bank with a Generation VI or VII game cartridge inserted. I was planning to start by transferring Pokémon from Gen V and had launched the app for the first time with a Gen V cartridge inserted.

Poké Transporter cannot be found by searching the eShop; it can only be downloaded from the Pokémon Bank menu while a Gen VI or Gen VII game is inserted.

  1. Insert a Gen VI/VII cartridge
  2. Launch Pokémon Bank
  3. Buy the 1-year pass
  4. Download Poké Transporter
  5. Quit Pokémon Bank
  6. Remove the game cartridge
  7. Insert any DS-era cartridge you want to transfer (skip this step if transferring from a Virtual Console game)
  8. Launch the game you want to transfer from, and move any Pokémon that you want to transfer into Box 1.
  9. Save and quit.
  10. Launch Poké Transporter
  11. Select the game from step 7
  12. Transfer the Pokémon from Box 1 into the Poké Transporter box
  13. Quit Poké Transporter
  14. Remove the DS-era cartridge (if applicable)
  15. Insert the 3DS-era cartridge
  16. Launch Pokémon Bank
  17. Move the Pokémon from the Poké Transporter box to another Pokémon Bank box, or into a box in your 3DS-era game.
  18. Repeat steps 6 - 17 until you've transferred all your Pokémon or are too exhausted to continue.