Do you have an id on the table? if so you do something like this:
$("table[id]").each(function(i,item){
var $table = $(item)
var table_id = $table.attr('id')
$table.find(".refresh-button").on("click", function() {
$table.load(location.href + " " + table_id);
})
})
This uses jquery to reload the page in an ajax call, and then swap the refreshed table into the document where the old data used to be.
ALTERNATELY:
you can define a new url, that just returns the rendered table, and call this in the inner function.
$table.load("/_tables/my_table/ " + table_id);