Call callback-function inside Behaviour-Module

Another option would be to generate all the required code into the module that says use. This replaces threading callback_module everywhere by local calls:

defmodule MyBehaviour do
  @callback specific_handling(data :: term) :: term

  @callback very_specific_handling(data :: term) :: term

  defmacro __using__(_opts) do
    quote do
      @behaviour MyBehaviour

      alias MyBehaviour

      def initiate_processing(data) do
        data
        |> do_stuff_with_data()
        # and other calls to behaviour-functions
        |> specific_handling()
      end

      defp do_stuff_with_data(data) do
        data
        # and other calls to behaviour-functions
        |> do_specific_stuff()
      end

      defp do_specific_stuff(data) do
        data
        # and other calls to behaviour-functions
        |> very_specific_handling()
      end
    end
  end
end

defmodule DataHandler11 do
  use MyBehaviour

  def specific_handling(data) do
    # do specific stuff with data
    data
  end

  def very_specific_handling(data) do
    # do specific stuff with data
    data
  end
end

defmodule DataHandler12 do
  use MyBehaviour

  def specific_handling(data) do
    # do specific stuff with data
    data
  end

  def very_specific_handling(data) do
    # do specific stuff with data
    data
  end
end

data = nil
handler = 1

case handler do
  1 -> DataHandler11.initiate_processing(data)
  2 -> DataHandler12.initiate_processing(data)
end

This has a cost: the code output by __using__ is compiled over again for every module that calls it, versus only being compiled once when MyBehaviour is compiled.