The GtkEntry already catches and consumes click events, so your controller won’t be notified.
You have to set the propagation phase of m_first_button_click_controller to GTK_PHASE_CAPTURE to be able to catch the events before the GtkEntry default controller does.
mouse_click = Gtk::GestureClick::create();
// This mouse event must be handled in the capture phase.
// The GtkButton C class handles the event in the capture phase and
// does not let it propagate to the bubble phase where events are
// handled by default.
mouse_click->set_propagation_phase(Gtk::PropagationPhase::CAPTURE);