Remove a primary site from multisite & replace with sub-site

I have a small 4 site multisite network. I've decided to migrate the primary site elsewhere as it's my own and I'd like to keep it separate from the client's sites.

It is a sub-directory multisite but they are all domain mapped. I'll protect my client's actual domains but essentially I have the following...

mysitedomain.co.uk - primary site
mysitedomain.co.uk/client-site-1 - mapped to clientdomain1.com
mysitedomain.co.uk/client-site-2 - mapped to clientdomain2.com
mysitedomain.co.uk/client-site-3 - mapped to clientdomain3.com

I've migrated the mysitedomain.co.uk site away and set it up as a standard non-multisite WordPress installation on a temporary URL.

Before I change the mysitedomain.co.uk DNS settings to point to the stand-alone install I think I need to promote one of the sub-sites to the primary site and give myself a setup like this...

clientdomain1.com - primary site
clientdomain1.com/client-site-2 - mapped to clientdomain2.com
clientdomain1.com/client-site-3 - mapped to clientdomain3.com
clientdomain1.com/mysite - unmapped

I could then archive or delete the unmapped ex-primary site from the multisite installation.

How is best to go about this and what's the order of operations? Or should I approach this a completely different way?

Thanks.