Proper template override of characteristic display

The documentation doesn't say that you should copy/paste that file.
It says that you need to create a hikashop_characteristics.php file and in it develop a function hikashop_characteristics_html(&$element,&$params,&$obj) and that to understand what you need to put in it, you'll want to look at the code in administrator/com_hikashop/types/characteristic.php
It actually won't be that easy.

In your case, I would do something different and not use the characteristic display override.
Instead, edit the file "show_block_characteristic" via the menu Display>Views, and change the line:
echo $this->characteristic->displayFE($this->element, $this->params) . '</div>';

to:
echo str_replace('return hikashopUpdateVariant(this);','return hikashopUpdateVariant(this);hikashopNoStockNoPrice();',$this->characteristic->displayFE($this->element, $this->params)) . '</div>';

That way, it's still done as an override, and even if you update HikaShop, you still use the standard characteristic system with just that JS function call added.