Magento 2: How to add button in system configuration?

I’m trying to add a button in the system configuration but also wanted to save it in the core_config_data. The reason is in every store view I want to show different messages based on the store id. But the problem is when I create a button in system.xml it does not save any path in core_config_data. Here is the code which I have done so far,

System.xml

<field id="sync_terminal" translate="label" type="button" sortOrder="7" showInDefault="1" showInWebsite="1" showInStore="1">
    <label>Synchronize payment methods</label>
    <frontend_model>VendorModuleBlockSystemConfigButton</frontend_model>
</field>

Controller file Button.php

<?php 

namespace VendorModuleControllerAdminhtmlSystemConfig;

use MagentoBackendAppAction;
use MagentoBackendAppActionContext;
use MagentoFrameworkControllerResultJson;
use MagentoFrameworkControllerResultJsonFactory;
use MagentoConfigModelResourceModelConfig;
use MagentoFrameworkAppConfigScopeConfigInterface;
use MagentoStoreModelScopeInterface;
use MagentoFrameworkAppResponseHttp as ResponseHttp;

class Button extends Action
{
    /**
     * Get country path
     */
    const COUNTRY_CODE_PATH = 'general/country/default';

    protected $resultJsonFactory;

    /**
     * @var SystemConfig
     */
    private $systemConfig;
    /**
     * @param Context $context
     * @param JsonFactory $resultJsonFactory
     */
    public function __construct(
        Context $context,
        JsonFactory $resultJsonFactory
    ) {
        $this->resultJsonFactory = $resultJsonFactory;
        parent::__construct($context);
    }

    /**
     * @return Json
     */
    public function execute()
    {

        $message = 'success';
        /** @var Json $result */
        $result = $this->resultJsonFactory->create();

        return $result->setData(['message' => $message]);
    }

}

Template file button.phtml

<script>
    require([
        'jquery',
        'prototype'
    ], function ($) {

        $('#btn_id').click(function () {
            var params = {};
            new Ajax.Request('<?php echo $block->getCustomUrl() ?>', {
                parameters: params,
                loaderArea: false,
                asynchronous: true,
                onCreate: function () {
                    $('#btn_id_response_message').text('');
                },
                onSuccess: function (transport) {
                    var resultText = '';
                    if (transport.status > 200) {
                        resultText = transport.statusText;
                    } else {
                        var response = JSON.parse(transport.responseText);
                        resultText = response.message
                    }
                    $('#btn_id_response_message').text(resultText);

                }
            });
        });
    });
</script>
<?php echo $block->getButtonHtml(); ?>
<p>
    <span id="btn_id_response_message"></span>
</p>

Block file Button.php

<?php
namespace VendorModuleBlockSystemConfig;

use MagentoConfigBlockSystemConfigFormField;
use MagentoBackendBlockTemplateContext;
use MagentoFrameworkDataFormElementAbstractElement;
class Button extends Field
{
    protected $_template = 'Vendor_Module::system/config/button.phtml';

    public function __construct(
        Context $context, 
        MagentoStoreModelStoreManagerInterface $storeManager,  
        array $data = []
        )
    {
        parent::__construct($context, $data);
        $this->_storeManager = $storeManager; 
        
    }
 
    protected function _getElementHtml(AbstractElement $element)
    {
        return $this->_toHtml();
    }

    public function getCustomUrl()
    {
        return $this->getUrl('vendormodule/system_config/button');
    }

        /**
     * @throws LocalizedException
     *
     * @return string
     */
    public function getButtonHtml()
    {
        $button = $this->getLayout()->createBlock(
            'MagentoBackendBlockWidgetButton'
        )->setData(
            [
                'id' => 'btn_id',
                'label' => __('Synchronize Terminals'),
                'class' => $this->getCurrentStoreId()
            ]
        );

        return $button->toHtml();
    }

    public function getCurrentStoreId()
    {
       /* Get Current Store ID */
       return $this->_storeManager->getStore()->getId();
    }
}