WebView Flutter: How to add bearer authorization header

The problem is the following:

1. I'm making authorization in my app and getting a token for bearer authorization.
2. After that, I need to create a WebView and pass the Authorization Bearer in the header

I've tried this, but in doesn't help.

import 'package:flutter/material.dart';

import 'package:skin_technology/core/consts/app_colors.dart';
import 'package:skin_technology/core/session_data/repo/session_data_service.dart';
import 'package:webview_flutter/webview_flutter.dart';

class CatalogWebView extends StatefulWidget {
  const CatalogWebView({Key? key}) : super(key: key);

  @override
  State<CatalogWebView> createState() => _CatalogWebViewState();
}

class _CatalogWebViewState extends State<CatalogWebView> {
  late WebViewController _webController;

  Future<String?> _getToken() async {
    final token = (await SessionDataService.sessionData)?.token;

    if (token == null) return null;
    return token;
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: const Text('КАТАЛОГ'),
        backgroundColor: AppColors.darkMainColor,
      ),
      body: FutureBuilder<String?>(
        future: _getToken(),
        builder: (BuildContext context, AsyncSnapshot snapshot) {
          switch (snapshot.connectionState) {
            case ConnectionState.waiting:
              return const Center(child: CircularProgressIndicator());

            case ConnectionState.done:
              if (snapshot.data == null) return Container();
              final headers = {"Authorization": "Bearer ${snapshot.data}"};
              const url = 'https://clubskt.ru/client/catalog';
              return WebView(
                javascriptMode: JavascriptMode.unrestricted,
                onWebViewCreated: (controller) async {
                  _webController = controller;
                  await _webController.loadUrl(url, headers: headers);


                },
                onPageStarted: (url) {},
              );

            default:
              return Container();
          }
        },
      ),
    );
  }
}

I tried the packages webview_flutter, flutter_webview_pro, flutter_webview_plugin etc. It still doesn't work.