I finally got Visual Studio Code’s terminal and tab handling working the way I wanted, with these keybindings:

[
  {
      "key": "shift+cmd+]",
      "command": "workbench.action.terminal.focusNext",
      "when": "terminalFocus"
  },
  {
      "key": "shift+cmd+[",
      "command": "workbench.action.terminal.focusPrevious",
      "when": "terminalFocus"
  },
  {
      "key": "cmd+w",
      "command": "workbench.action.terminal.kill",
      "when": "terminalFocus"
  },
  {
      "key": "ctrl+`",
      "command": "workbench.action.terminal.focus"
  },
  {
      "key": "ctrl+`",
      "command": "workbench.action.focusActiveEditorGroup",
      "when": "terminalFocus"
  }

]

This way, cmd+shift+] and cmd+shift+[ not only allow you to switch between editor tabs, but also terminals, when the terminal is focused. cmd+w kills the current terminal tab as expected. ctrl+` switches focus between the editor and the terminal.

I was considering overriding cmd+t to open a new terminal when the terminal is focused, but decided to just get used to the built in ctrl+shift+` shortcut instead.