Splitting an eCommerce store from the WordPress Theme

Core site functionality really shouldn’t be so tightly coupled to the theme. This is why the plugin system exists. If it’s not too late I would recommend exploring alternatives.

That said, anything happening on the admin side is in the theme’s functions.php file, or is included via that file. As long as that file and anything it loads is in the proper place it’s likely the admin side will work. Then it’s a matter of merging or including the other templates into your own theme to get the presentation you want on the front end.