I use the following script (inside a controller – action for now) for duplicating a product programatically.

public function createAction(){
    $data = $this->getRequest()->getParams();

    $product = Mage::getModel('catalog/product');
    $_product = $product->load($data['prod_id']);

    $clone = $_product->duplicate();
    $clone->setSku($data['dup_prod_sku']);
    // $clone->setVisibility(Mage_Catalog_Model_Product_Visibility::VISIBILITY_BOTH);

    $clone->setPrice($data['dup_prod_price']);
    $clone->setSpecialPrice($data['dup_prod_sp_price']);
    $manage_stock = $data['dup_manage_stock'];
    $qty = ($data['dup_prod_stock'] == "") ? 0 : trim($data['dup_prod_stock']);
    $is_in_stock = $data['dup_is_in_stock'];
    if($manage_stock == 1){
        $stockArray = array(
                        'use_config_manage_stock' => 0,
                        'manage_stock' => 1,
                        'qty' => $qty,
                        'is_in_stock' => $is_in_stock,
                    );
    } else{
        $stockArray = array(
                        'use_config_manage_stock' => 0,
                        'manage_stock' => 0,
                    );
    }
    $clone->setStatus(1);
    $_idArray = Mage::helper('marketplace/vendor')->getVendorIdFromUserId();
    $_vendor = Mage::getModel('marketplace/vendorcode')->getCollection()
                            ->addFieldToSelect('attr_opt_id')
                            ->addFieldToFilter('vend_user_id', array('eq' => $_idArray['user_id']))
                            ->getFirstItem();
    $clone->setVendor($_vendor->getAttrOptId());
    $clone->setTaxClassId(4);
    $image_url = $this->getBaseImagePath($_product->getImage());
    $clone->setMediaGallery(array());
    $clone->setStockData($stockArray);
    try{
        $clone->getResource()->save($clone);
        $new_product = $product->load($clone->getId());
        $new_product->addImageToMediaGallery($image_url, array ('image', 'small_image', 'thumbnail'), false, false);
        $new_product->save();
        $stockItem = Mage::getModel('cataloginventory/stock_item')->loadByProduct($clone->getId());
        foreach($stockArray as $key => $val){
            $stockItem->setData($key, $val);
        }
        $stockItem->save();
    } catch(Exception $e){
        Mage::log($e->getMessage());
    }
    $this->_redirect('*/*/productgrid');
}

This works well and the product gets duplicated with supplied SKU and overwritten prices from a form.

• I can see the product in product grid in admin panel.
• Visibility is set to Catalog, Search
• Product is in stock
• Enabled and tagged to correct category and website.

But still I cant get it to display in the category page in the frontend, whereas I can see the products duplicated from admin panel without any problem.

Moreover If I try to reindex, Product Flat Data isn't getting reindexed and is throwing SQL foreign key constraint error.

NOTE:
Vendor is a custom attribute