Can't find a staging site solution that retains an existing site

I’ve read everything I can find on this subject for the past week, but haven’t found an answer. My situation is that I currently have a website up and running that is NOT in WordPress. I’m building a new site in WP but I cannot take down the current site and put up an “under construction” page. I need that site to remain where it is until my new WordPress version is finished. So…

I have installed WordPress in the root directory if the same site. I don’t need the new site to have the same URLs as the old ones. I would like to create a “staging” site right where the old one is so I don’t have to re-create the WP installation when I’m finished (not quick and not easy). In order to do that I need to have WordPress read mysite.com/home/ as my WP home page – and leave mysite.com/ undisturbed.

I tried the Disable Canonical URL Redirects plugin. Before activating it mysite.com/home/ displayed a crashing version of the current live home page with no scripts or images running. After activation it went to my server’s 404 page and has remained there ever since, whether the plugin is on or off (or .htaccess contains redirects or not).

I also tried this solution WordPress Staging Environment In 5 Quick and Dirty Steps which appears to be a unique solution but the redirect from my staging subdomain takes me to this when logged into the CPanel or the WP login page when I’m logged out – not my site’s staging theme’s home page.

Given that staging sites are a development necessity in many cases these days and that all of the WP code is accessible, there MUST be a way to do this. Any ideas?

Thanks Charles