こちらとほぼ同じですが、scopesに'https://www.googleapis.com/auth/cloud-platform'を追加しないと403エラーが帰ってきて詰まってたので備忘録です

void main() async {
  var json = {};
  var credential = await Credentials.fetch();
  var fbClient = FirebaseClient(credential);
  //データ.jsonに書き込み
  await fbClient.post(
     'https://firebasestorage.googleapis.com/v0/b/[ProjectID].appspot.com/o/データ.json',
      json);
}

class Credentials {
  static Future<String> fetch() async {
    var pk = await getPrivateKey();

    var accountCredentials = ServiceAccountCredentials.fromJson({
      'private_key_id': pk['private_key_id'],
      'private_key': pk['private_key'],
      'client_email': pk['client_email'],
      'client_id': pk['client_id'],
      'type': 'service_account',
    });

    var scopes = [
      'https://www.googleapis.com/auth/datastore',
      'https://www.googleapis.com/auth/cloud-platform'//stackoverflowとかだとこれはなかったけど、ないと権限が得られず403エラーを吐いてしまう
    ];

    var client = http.Client();
    var credentials = await obtainAccessCredentialsViaServiceAccount(
        accountCredentials, scopes, client);
    client.close();
    return credentials.accessToken.data;
  }

  static Future<Map<String, dynamic>> getPrivateKey() async {
    String jsonString =
        await File('/Path/to/firebase_private_key/keyfile.json')
            .readAsString();
    return json.decode(jsonString);
  }
}