Need Advice – Install WP on client's servers but keep old site live until switch

Hi, I’m looking for advice how to install this.

Currently an existing client’s website eg. https://www.clientsite.com, that needs to stay live until I switch to new wordpress site.

https://www.clientsite.com is on client’s hosting plan and staying there.
I need a few weeks to install/style/build up content on new wordpress site before I can make the switch. What would be the best workflow??

1. Install wordpress on a subdomain on my own hosting plan to build site eg. https://www.wordpressinstall.mypersonalsite.com
2. Install WP database on subdomain on their hosting plan eg. https://www.wordpressdatabase.clientsite.com
3. Build up wordpress site on my servers, then copy all wordpress install files to their servers, thus replacing the actual https://www.clientsite.com, and hope all the links and everything work?

If anyone has any experience with this workflow, please advise me!!
Thanks in advance,
Stephen