Pause status timer while modals are open (#3131)

Summary:
- pause the status timer while waiting on approval modals
- expose deterministic pause/resume helpers to avoid sleep-based tests
- simplify bottom pane timer handling now that the widget owns the clock

codex-rs/tui/src/bottom_pane/mod.rs

@@ -162,6 +162,8 @@ impl BottomPane {
             view.handle_key_event(self, key_event);
             if !view.is_complete() {
                 self.active_view = Some(view);
+            } else {
+                self.on_active_view_complete();
             }
             self.request_redraw();
             InputResult::None
@@ -201,6 +203,8 @@ impl BottomPane {
             CancellationEvent::Handled => {
                 if !view.is_complete() {
                     self.active_view = Some(view);
+                } else {
+                    self.on_active_view_complete();
                 }
                 self.show_ctrl_c_quit_hint();
             }
@@ -381,10 +385,27 @@ impl BottomPane {
 
         // Otherwise create a new approval modal overlay.
         let modal = ApprovalModalView::new(request, self.app_event_tx.clone());
+        self.pause_status_timer_for_modal();
         self.active_view = Some(Box::new(modal));
         self.request_redraw()
     }
 
+    fn on_active_view_complete(&mut self) {
+        self.resume_status_timer_after_modal();
+    }
+
+    fn pause_status_timer_for_modal(&mut self) {
+        if let Some(status) = self.status.as_mut() {
+            status.pause_timer();
+        }
+    }
+
+    fn resume_status_timer_after_modal(&mut self) {
+        if let Some(status) = self.status.as_mut() {
+            status.resume_timer();
+        }
+    }
+
     /// Height (terminal rows) required by the current bottom pane.
     pub(crate) fn request_redraw(&self) {
         self.frame_requester.schedule_frame();