Flutter Web: Http request is unable to send my header named 'Cookie', is there a way to fix this?

I am calling my API which require a header named 'Cookie' for user auth. I'm using the Http package to make the call and everything is working fine in my Android and iOS app. Now I'm testing my app on web but my API calls are failing. Getting this error -> Refused to set unsafe header "Cookie"

This is the code:

Future<UserData> getUserData() async {
    try {
      final response = await http.get(
        Uri.parse(_baseUrl),
        headers: {
          'Content-type': 'application/json; charset=utf-8',
          'Cookie': _authHeader,
        },
      );

      if (response.statusCode == 200) {
        return UserData.fromJson(response.body);
      } else {
        return Failure(message: 'Failed to get data!');
      }
    } catch (e) {
      return Failure(message: e.toString());
    }
  }

When I run this code on Android or iOS it works perfectly but when I run it on Web the browser does not send the header named 'Cookie' and the call fails. This happens on localhost as well as in the production which is hosted on a secured domain.

I tried some solutions but nothing worked, I have also updated the CORS policies on my server and allowed my localhost port and my domain there, still no change.

On my flutter code I tested this but no difference

Future<UserData> getUserData() async {
    try {
      final webClient = (http.Client() as BrowserClient)
        ..withCredentials = true;

      final response = await webClient.get(
        Uri.parse(_baseUrl),
        headers: {
          'Content-type': 'application/json; charset=utf-8',
          'Cookie': _authHeader,
        },
      );

      if (response.statusCode == 200) {
        return UserData.fromJson(response.body);
      } else {
        return Failure(message: 'Failed to get data!');
      }
    } catch (e) {
      return Failure(message: e.toString());
    }
  }