Changing primary domain's path on VPS

I'm assisting with a site rebuild for an organization. Currently they have the new site on a secondary hostgator account. The goal is to minimize downtime and minimize any deletions.

So my thought was "put the new site in a subdirectory on the old site, get everything lined up, and switch your domain's binding from /public_html/ to /public_html/newsite/".

Unfortunately this is not an addon domain. I went in to my personal cpanel to demonstrate how it would be done and noticed that I can't easily change my primary domain's mapping from within cpanel.

What should I do? Would renaming public_html to public_htmlold and creating a new public_html directory and dropping the new site in to it break anything?

Thanks!