Custom Properties Layout limitation

We're developing a site that relies on many Custom Property Layout menu items to categorize and display a manufacturer's products. Unfortunately we just discovered that we have no way of controlling the appearance (module assignment, etc.) of the page that follows clicking on an item in the Custom Property Layout page.

In other words, a Custom Property Layout menu item has been assigned a page class of "automobile" to display a certain segment of automobiles. That page displays correctly and all is well, UNTIL you click on one of the selections displayed on that page. The new page (article) opens up with a page class of "default" and applies all the default page styling and assigned modules. There doesn't seem to be any way to alter that behavior.

Ideally, the click-through pages should have the same page classes as those that are assigned to the Custom Property Layout menu item. Without that functionality the component won't be at all useable for us.

Am I missing something obvious somewhere? Thanks --

The content items retrieved by Custom Properties need to have their "organic" menu item to open under.
When none exists, the content item opens under a generic page.

It all comes down to the way Joomla builds SEF routes for content items.
A short example to illustrate the concept:
You need to build the route (SEF link) for article.
Joomla looks within the *menu items* if there's one that points exactly to that article.
If so, the route to the article is that menu item.
If there's no exact match, Joomla looks for menu items pointing to the category or ancestor category of that content items.
If found, the route to the article is the menu item of its parent category + article alias
If no menu items points to the content item or to its category, then there's no route fo the article; this implies that you have no place where you can control the page appearance (module assignemnt, page class, etc) because those settings are controlled by menu item parameters.

So, you need to have one or more Joomla articles and/or category and/or blog menu items that your content items "organically" belongs to. Said menu items could also be hidden ones.

To use your example, and assuming all products belongs to one category, I'd add a Joomla Article Blog View named "parts", showing all "parts" items.
You can then proceed to customize this default "landing" page, by editing the page class, adding modules, etc, etc.

That should fix it.