In Squarespace terms, those are two distinct sites. They cannot both point to the same domain.
However, if I understand correctly, you don't need the second site. Build your new site as unlinked pages in Design A.
For example, if your main page is called Home, create a new page called Home-New. Under settings, mark it as inactive so it's not visible to the world (or password-protect it).
Once the new design is ready, simply rename Home to Home-Old, rename Home-New to Home and mark it as your home page.
Do the same for whatever other pages need to change.
Your domain will be unaffected.
