How do I add a relation parameter to my filter query?

Each of your code blocks will overwrite the previous, you need to add to the array rather than replace it:

$args['tax_query'][] = array(
    'taxonomy' => 'region',
    'field'    => 'id',
    'terms'    => $_POST['region_filter']
);

relation is AND by default, so you can just omit it, but if you wanted to explicitly set it:

$args['tax_query']['relation'] = 'AND';