Hi,
We would need to see the page with the issue to be able to say more.
Right now, without the code modified and with the form only available if you already purchased the product, we can't see anything.
If you can't do it on your live website, then try making a copy of the website so that we can look at the issue on the copy.
Also, I would recommend something simpler like:
<input class="button btn" type="button" value="<?php echo JText::_('HIKASHOP_SEND_COMMENT'); ?>" onclick="hikashop_send_comment(); alert('You\'re review has been received and will soon be approved by a moderator. Thank you!');"/>
Because with your code, if another extension has a javascript function named success, it will break the extension.
That's why everything you see in the code of HikaShop is prefixed with hikashop so that there is no conflict with something else.