Visual composer is a page builder.
If the other site does not use VC then in the interests of less stress and avoiding issues with styling etc. You are better off trying to build it up from scratch using VC in your main theme.
If there are specific features that oh want from the existing site that don’t exist by default in the new one then you will probably have to custom create them.
Have you got links to the page you are trying to replicate and BOTH theme names?