In case someone still needs a better answer, I would like to add mine.
First, I created a Splash Screen Activity which first connects to the byethost server and get the response and then parse the "__test" cookie from it.
void getCookie() {
    RequestQueue mQueue = Volley.newRequestQueue(this);
    StringRequest stringRequest = new StringRequest(Request.Method.GET, Constants.SERVER_URL,
            response -> {
                try {
                    if (response.contains("src=\"/aes.js\"") || response.contains("src=\"/aes.min.js\"")) {
                        String beginOffsetA = "var a=toNumbers(\"";
                        String beginOffsetB = "\"),b=toNumbers(\"";
                        String beginOffsetC = "\"),c=toNumbers(\"";
                        String endOffsetC = "\");document.cookie=";
                        String a = response.substring((response.indexOf(beginOffsetA) + (beginOffsetA).length()), response.indexOf(beginOffsetB)); // Value of var a
                        String b = response.substring((response.indexOf(beginOffsetB) + (beginOffsetB).length()), response.indexOf(beginOffsetC)); // Value of var b
                        String c = response.substring((response.indexOf(beginOffsetC) + (beginOffsetC).length()), response.indexOf(endOffsetC)); // Value of var c
                        Constants.COOKIE = "__test=" + encrypt(hexStringToByteArray(a), hexStringToByteArray(b), hexStringToByteArray(c)).toLowerCase() + "; expires=Thu, 31-Dec-37 23:55:55 GMT; path=/"; //This is the "__test" Cookie, e.g., "__test=8927389y1huwieqyue"    
                    } else {
                        theServerDoesNotNeedTestCookie();
                    }
                } catch (Exception e){
                    e.printStackTrace();
                    didntWork();
                }
            },
            error -> doesNotWork();
    );
    mQueue.add(stringRequest);
}
public String encrypt(byte[] key, byte[] initVector, byte[] data) {
    try {
        IvParameterSpec iv = new IvParameterSpec(initVector);
        SecretKeySpec keySpec = new SecretKeySpec(key, "AES");
        Cipher cipher = Cipher.getInstance("AES/CBC/NoPadding");
        cipher.init(Cipher.DECRYPT_MODE, keySpec, iv);
        byte[] encrypted = cipher.doFinal(data);
        return bytesToHex(encrypted);
    } catch (Exception ex) {
        new Reporter(this, ex);
    }
    return null;
}
public String bytesToHex(byte[] bytes) {
    final char[] HEX_ARRAY = "0123456789ABCDEF".toCharArray();
    char[] hexChars = new char[bytes.length * 2];
    for (int j = 0; j < bytes.length; j++) {
        int v = bytes[j] & 0xFF;
        hexChars[j * 2] = HEX_ARRAY[v >>> 4];
        hexChars[j * 2 + 1] = HEX_ARRAY[v & 0x0F];
    }
    return new String(hexChars);
}
public byte[] hexStringToByteArray(String s) {
    int len = s.length();
    byte[] data = new byte[len / 2];
    for (int i = 0; i < len; i += 2) {
        data[i / 2] = (byte) ((Character.digit(s.charAt(i), 16) << 4)
                + Character.digit(s.charAt(i+1), 16));
    }
    return data;
}
This works for byethost.