Integrating a PHP script & database with theme

Hello, I would like to run a PHP script with a mySQL database while using my wordpress theme. The script calls from a separate database from the one wordpress uses, and runs a pagination as well as categories, and I haven’t found a way to integrate it using ‘pages’ in wordpress.

There is a solution that I am contemplating: create a file under my theme’s folder to use the header.php, etc. But that would mean that I’d have to use the full url of that file (https://site.com/wordpress/wp-content/themes/themename/file.php) to show my visitors and I don’t want to do this.

Is there a way to integrate my script, by either masking the file’s name, or does anyone have a different solution?