要解决AutomaticKeepAliveClientMixin在BottomNavigationBar页面中不起作用的问题，可以考虑使用AutomaticKeepAliveClientMixin的子类KeepAliveWidget来包装BottomNavigationBar。以下是一个示例代码：

import 'package:flutter/material.dart';

class MyBottomNavigationBar extends StatefulWidget {
  @override
  _MyBottomNavigationBarState createState() => _MyBottomNavigationBarState();
}

class _MyBottomNavigationBarState extends State with AutomaticKeepAliveClientMixin{
  int _currentIndex = 0;

  @override
  bool get wantKeepAlive => true;

  @override
  Widget build(BuildContext context) {
    super.build(context); // 这一行很重要，必须调用父类的build方法

    return Scaffold(
      body: _getPageByIndex(_currentIndex),
      bottomNavigationBar: BottomNavigationBar(
        currentIndex: _currentIndex,
        onTap: (int index) {
          setState(() {
            _currentIndex = index;
          });
        },
        items: [
          BottomNavigationBarItem(
            icon: Icon(Icons.home),
            title: Text('首页'),
          ),
          BottomNavigationBarItem(
            icon: Icon(Icons.settings),
            title: Text('设置'),
          ),
        ],
      ),
    );
  }

  Widget _getPageByIndex(int index) {
    switch (index) {
      case 0:
        return HomeScreen();
      case 1:
        return SettingsScreen();
      default:
        return HomeScreen();
    }
  }
}

class HomeScreen extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Container(
      child: Text('首页'),
    );
  }
}

class SettingsScreen extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Container(
      child: Text('设置'),
    );
  }
}

void main() {
  runApp(MaterialApp(
    home: MyBottomNavigationBar(),
  ));
}

在这个示例中，通过在MyBottomNavigationBar的状态类中混合AutomaticKeepAliveClientMixin，并重写wantKeepAlive方法来实现自动保持页面状态。同时，还需要在build方法中调用super.build(context)来确保父类的build方法被调用。这样就可以保持BottomNavigationBar中的页面状态了。