Do you have any hints about how I would write some code within this custom page template that would call on another theme’s files…
You don’t call another theme’s files – you copy the relevant code from that theme’s files into your custom page. You also have to copy the relevant CSS. Depending on how different the styling is, you can either include it in your main stylesheet at the bottom, or put it in its own stylesheet and conditionally include that stylesheet in the main theme’s header.php file when the user is viewing the special page.
What I would do is assign a class to the “content” div on the custom page template. As example, if the page is called “Special”, define content
<div id="content" class="special">
=== code goes here ===
</div>
then in your css you can do things like…
if the content div on normal pages is defined as
#content { background-color: #ffffff; }
you can apply a different background color on the special page
#content.special { background-color: #e1e1e1; }
in other words you overwrite the default styling assigned to other pages. To make the p tags on that page be a different font size than all other pages:
#content.special p { font-size: 14px; }
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If you’re not a coding type and this sounds too complicated, another approach is to use two separate WordPress installations (install the second in a folder). Install the the theme you want on the 2nd site, duplicate the main site’s nav, and interlink that page with the main site in the nav on both sites.