Magento – Cart template $this->getItems() as $_item obtains configurable details not simple

You need to find the simple product in the cart that is associated to the configurable product and get the attribute from that one.
The simple cart item is linked through the configurable one through $item->getParentItemId(). So you should loop through all the cart items, not just visible ones until you find one that matches your needs.
Here is a possible way of doing it.

protected function _checkCarrierAvailability($carrierCode, $request = null)
{
    foreach (Mage::getSingleton('checkout/cart')->getQuote()->getAllVisibleItems() as $item)
    {
        if($item->getProduct()->isConfigurable()) {
            //loop through all products
            foreach (Mage::getSingleton('checkout/cart')->getQuote()->getAllItems() as $cartItem) {
               if ($cartItem->getParentItemId() == $item->getId()) {
                   //you found the simple product
                   if($item->getProduct()->getOverweight()) {
                        if($carrierCode == 'flatrate'){
                            return false;
                        }
                   }
                   break; //no need to search further
               }
           }
        } else {
            if($item->getProduct()->getOverweight()) {
                if($carrierCode == 'flatrate'){
                    return false;
                }
            }
        }   
    }
    return true;
}

What I described above can be used in the general case, when you need something from the simple product, but in your case is even more simple.
All you need is to find a product that is overweight. If you have one it means you cannot use flatrate.
You shouldn't care if the product is configurable or not. You just need one item. So instead of looping through getAllVisibleItems you should loop through getAllItems

protected function _checkCarrierAvailability($carrierCode, $request = null)
{
    foreach (Mage::getSingleton('checkout/cart')->getQuote()->getAllItems() as $item)
    {
        if($item->getProduct()->getOverweight()) {
            if($carrierCode == 'flatrate'){
                return false;
            }
        }
    }   
    return true;
}

The issue I was running into was that loadByAttribute() does not act in the same way that load() does and therefore I would need to fetch the id of the simple product then once again load the simple product. Below is the corrected code:

$productId = $_item->getProductId();
$product = Mage::getModel('catalog/product')->load($productId);
if($product->getTypeId() == "configurable"){
    $sku = $_item->getSku();
    $simpProdId = Mage::getModel('catalog/product')->loadByAttribute('sku',$sku)->getId();
    $product = Mage::getModel('catalog/product')->load($simpProdId);
}
$invStatus = $product->getResource()->getAttribute('product_inventory_status')->getFrontend()->getValue($product);
$quantity = Mage::getModel('cataloginventory/stock_item')->loadByProduct($product)->getQty();