Fix authentication when used in a Plugin context

[jacobbudin]

App SDK's SanityApp does not authenticate in a Plugin (Tool) context.

To Reproduce

1. Implement App SDK into an existing Studio as a Tool, e.g.:

const MyComponent: FunctionComponent = (): ReactElement => {
  const client = useClient()
  const {projectId, dataset} = client.config()
  const config: SanityConfig[] = [
    {
      projectId,
      dataset,
    },
  ]

  return (
    <SanityApp
      config={config}
    >
      <MySubComponent />
    </SanityApp>
  );
}

export const myTool = (): Tool => {
  return {
    component: () => <MyComponent />,
  };
}

2. Try loading the Tool in any environment other than localhost (https://example.com)
3. Wait for 5 seconds, the Tool will disappear and be replaced with an authentication error

Observe CORS-related failed network requests to:

• https://api.sanity.io/v2021-06-07/users/me
• https://api.sanity.io/v2025-02-19/projects/{project_id}?tag=sanity.sdk

Expected behavior
Authentication should persist and succeed. The Tool should load as it does on localhost.

Screenshots

Which versions of Sanity are you using?
v3.93.0

What operating system are you using?
Not applicable

Which versions of Node.js / npm are you running?
Not applicable

See associated Sanity Discord thread.