Yes it's always possible. THe only thing you have to do is to modify the base template to add the status tag in the generaated url by concatenation it should work.The it would be sent like any other get param that you can retrieve.
If the final url isn't called in a template but in an extra elastic block or controller file, the same logic applies. All you have to do is to find where the link is called.
Assuming your url is built in there and assuming you want to do it for every request
here is the method you need to extend from vendor/magento/module-search/Helper/Data.php
/**
* Retrieve result page url and set "secure" param to avoid confirm
* message when we submit form from secure page to unsecure
*
* @param string $query
* @return string
*/
public function getResultUrl($query = null)
{
return $this->_getUrl(
'catalogsearch/result',
['_query' => [QueryFactory::QUERY_VAR_NAME => $query], '_secure' => $this->_request->isSecure()]
);
}
All you have to do is to make a preference on that class, and to rewrite the method concatenating on the return ."&status=".$status
Second argument of getUrl is also taking params, may be also check if you cannot set the status directly in there rather than concatenating which is kinda ugly way to do it.