maybe ....

foreach ($_POST as $k=>$v){$posted[$k] = $v;}

$invoice_number = $posted['invoice'];

$invoice_shipping_query = tep_db_query("SELECT value FROM orders_total WHERE orders_id = '" . $invoice_number . "' and class = 'ot_shipping'");
$invoice_shipping = tep_db_fetch_array($invoice_shipping_query);
$shipping_cost = $invoice_shipping['value'];

$posted['shipping'] = $shipping_cost;

that was a great try. didnt work.

First does array_push_assoc() function exist? Its not a built in PHP function.

You could just use PHP's array_push()
array_push( $shipping_cost, 'shipping');

print_r( $shipping_cost);

Will now have shipping added to the array.

NOTE: do not assign array_push to a var, its a converter and the assigned var will output a count not the array.

To push key => value you would need a function something like this.
function array_push_assoc($array, $key, $value){
$array[$key] = $value;
return $array;
}

This pushes the keys also.
But the function you are using is using different params
In which your first param should result in an undefined variable.
So I am not sure what you are trying to do there.

Now is it working fine?. Let us know