Custom *Page* Types

I’m wondering if there’s a way to activate the Page Attributes -> Template drop menu seen on Pages (not Posts) when using Custom Post Types.

In other words, I want to everything that pages does but be able to create multiple custom labeled page sections in the admin, including matching blocks in the Appearance -> Menus.

TRIED
I found the 'capability_type' => 'page', setting for registering a post type in the functions.php file, but that seems to be about permissions and doesn’t work.

USAGE
A restaurant has 2 locations with distinct but similar page needs for each. For example, both locations need separate About, Contact, Dinner Menu, Lunch Menu, etc. pages. The two locations cannot share these pages because the info is different for each. If I put all of these into the one standard Pages section, there’s no easy way to distinguish between what looks like duplicate pages. Same is true for the Appearance -> Menus section.

ALTERNATIVES
I know people ask why I’d want to select page templates when I can create a custom template via template files (single-[templatename].php or whatever). However, that’s not as flexible. I have multiple page templates already available I’ll need to select for various custom pages I want to build.

I also don’t need special taxonomies or category selections such as with examples like music review blogs that might want artist, album, genre, etc. designations. I don’t need that, only pages functionality that are custom labeled and menus.