chore: Use the sink of the specific async logger directly for reporting the dropped messages

include/spdlog/async_logger-inl.h

@@ -62,6 +62,16 @@ SPDLOG_INLINE void spdlog::async_logger::backend_sink_it_(const details::log_msg
             SPDLOG_TRY
             {
                 sink->log(msg);
+
+                if (auto pool_ptr = thread_pool_.lock())
+                {
+                    auto lost_messages = pool_ptr->overrun_counter();
+                    if (lost_messages > 0) {
+                        sink->log(details::log_msg{name(), level::warn, fmt::format("Lost {} messages", lost_messages)});
+                        sink->flush();
+                        pool_ptr->reset_overrun_counter();
+                    }
+                }
             }
             SPDLOG_LOGGER_CATCH(msg.source)
         }
@@ -71,15 +81,6 @@ SPDLOG_INLINE void spdlog::async_logger::backend_sink_it_(const details::log_msg
     {
         backend_flush_();
     }
-
-    if (auto pool_ptr = thread_pool_.lock())
-    {
-        auto lost_messages = pool_ptr->overrun_counter();
-        if (lost_messages > 0) {
-            err_handler_(fmt::format("Lost {} messages.", lost_messages));
-            pool_ptr->reset_overrun_counter();
-        }
-    }
 }
 
 SPDLOG_INLINE void spdlog::async_logger::backend_flush_()