Output items from XML feed when matching to category - php

Question

I've exported all product items from my shop as an XML-File and parsed these items with PHP in my single.php of my theme. The parse works fine so far.
Now i want to only display products which category name matches with the ones that have been set in the article editor of WP. This is my code:
<?php
// Parse XML Feed
$html = "";
$url = "http://example.com/feed.xml";
$xml = simplexml_load_file($url);
$item = $xml->channel->item;
$categories = get_the_category();
for($i = 0; $i < count($item); $i++){
$title = $item[$i]->title;
$link = $item[$i]->link;
$category = $item[$i]->category;
$img = $item[$i]->description->a->img["src"];
// This approach breaks my site. A loop in another loop seems not to be that good
// foreach ($categories as $i => $cat) {
// $catName = $cat->name;
// if (preg_match("/\b$catName\b/", $category)) {
// $html .= "
// <a href='$link'><h3>$title</h3>
// <img src='$img'>
// <p>$category</p></a>
// <hr>";
// }
// }
if (preg_match("/\bCATEGORY_NAME\b/", $category)) {
$html .= "
<a href='$link'><h3>$title</h3>
<img src='$img'>
<p>$category</p></a>
<hr>";
}
}
echo $html;
?>
If i change CATEGORY_NAME to an category name which exits, then the output works like i want. The other approach with that loop breaks my site with an "Fatal error: Allowed memory size of *** bytes exhausted…".
Would love to hear some tips
EDIT: Here is some example code
<?xml version="1.0" encoding="UTF-8"?>
<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
<channel>
<atom:link href="www.example.com/export.xml" rel="self" type="application/rss+xml" />
<title>www.example.com</title>
<description>desc</description>
<link>www.example.com</link>
<language>1-1</language>
<image>
<url>www.example.com/logo.gif</url>
<title>test page</title>
<link>http://example.com</link>
</image>
<item>
<title>Example item</title>
<guid>www.example.com/item</guid>
<link>www.example.com/item</link>
<description>
<a href="www.example.com/item" style="border:0 none;">
<img src="www.example.com/image.png" align="right" style="padding: 0pt 0pt 12px 12px; float: right;" />
</a>
</description>
<category>Categoryname</category>
<pubDate>Thu, 24 Aug 2017 17:19:17 +0200</pubDate>rn</item>
</channel>
</rss>