How would you test that?

Following dependency rejection you would design the system to obtain the initial state before even attempting to start the GenServer - then there is no dependency that needs to be mocked out because the initial state is handed in as a plain value.

Two things come to mind:
Unless you want the whole application to fail at startup if the initial state request fails, issue an Agent.cast after the start_link to initialize the Agent. Let the cast handler do the init. Since that cast will be first in the message queue, nothing will get to the Agent before it’s initialized. Also, in the init, you can use an Application env to control if it pulls that initial state. While you can’t use Mix.env, you can use a config that’s dependent on the Mix.env that’s set up at compile/release.

How about just not testing the genserver started by your supervision tree, but starting up additional ones with custom configuration just for your tests:

setup do
  {:ok, pid} = start_supervised({MyApp.TheWorker, mock_config()})
  {:ok, pid: pid}
end

test "whatever", %{pid: pid} do
  assert MyApp.TheWorker.do_stuff(pid)
end