new multisite to replace existing static site

First time doing this, so appreciate any and all help…

How can I move my new WP Multisite network into place on live server with the existing website still running (to avoid downtime)?

Website name is same as the primary domain, i.e. ‘mywebsite.com’ is the primary domain account and simultaneously the domain name for the website. It will also serve as the domain name for the new WP Multisite network, which at present is using ‘mywebsite.dev’.

Lerner Consulting: “WordPress 3.0 Multi-Site Installation with Existing Web Sites” says to copy/move the existing website into an addon domain, having pointed the nameservers to the addon domain and use a placeholder domain name for the primary domain, where the new website will be installed at root. After new website is ready to go, switch the name servers back to primary domain.

But my webhost (shared hosting) is suggesting instead to create a subdomain and send the website from local to live server into that subdomain, test it, and then move it into the main folder.

That ought to work fine for a normal WP installation, but mine is a Multisite network, and requires to be installed into public_html.

I did successfully install WP Multisite network into an addon domain on the same account the other day, and it is running fine. But this other WP Multisite network needs to use the primary domain name.

Please can anyone here advise me what is the best and most straightforward way to get my new website online without downtime or with minimal downtime of the existing website?