Hi. Looks like you started not from the right side. The menu is a consequence, but not the essence. And of course it is not related to the template and especially to the PAGE Builder.
You can achieve this in two ways.
Create multiple sites with the same branding and menus, place them on a subdomains and relink their menus. This is handmade.
If you need the single authorization on all sites and control the menu from one admin, you will need to create a cluster of sites. For this purpose, use extensions like this. alterbrains.com /joomla-extensions/administration/multisites-manager