There are a few tips and tricks to improve the performance, please check the chapter 18.1 from the documentation.
If the default search bar is not replaced, it means that the theme is not using the “searchform.php” file to display the form, thus it's only replaceable by editing the theme code.
To help you, I have logged in via the FTP details and looked for the search box code to replace. I have found it in the wp-content/themes/Couponize/header.php file on lines 115-148.
I have placed the search shortcode to line 113 in that file, in case you want to edit it. I have also soft-disabled the old search form by adding html comment tags to line 114 and 148.
If you want to re-enabled the old search, just remove the ajax search pro shortcode from line 113 and the HTML shortcode tags from line 114 and 148.Best,
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