Unable to add and update product rows in the admin interface for an order

The admin order edit page allows administrators to add products to existing orders and to remove them. However, when you add or remove a product, all products are duplicated.

I did some digging and discovered the cause was that product fields marked as hidden do not end up in the product form. This is a problem because uc_order_product_save inserts new entries for every product in $_POST for which order_product_id is not set. And since order_product_id is a hidden field, it does not end up in $_POST causing every product to be inserted rather than updated.

The best, quickest and least painful solution I could come up with was to move the hidden fields for every row in the product form underneath a field that is visible, as in the attached patch.

However, if there's a better solution, and one that does not involve modifying core Ubercart core, I'd love to hear about it so I can implement that instead.