I would make a plugin out of the code that you are using to pull the XML. I’ve done this before for a client that was using a calendaring system that was a set of PHP files. I wrapped the calls to the calendar files in a plugin, and just called what I needed to from there. This preserved the look and feel of the site without extra work there, and used WordPress’s native capabilities.
It’s fairly easy to take an existing set of code and wrap it as a plugin, and then you can use the methods of that plugin in your theme files. You could for example make a page to house the data, make a page-specific theme file that outputs (or not) the content of that page, and also use the theme file to run the methods for pulling in the XML data via your plugin.
Resources for writing (or wrapping code into) a plugin: