How to Design a new WordPress site while keeping current WordPress site intact

You should clone their site onto a localhost installation, work on it, then show the client for approval (either on your local computer or on a sandbox like https://wpsandbox.net/). Then use a migration plugin to overwrite the old site, or just empty the database and load the new one.

If you don’t like that, you can install WordPress into a folder on their site, tick the box for discouraging search engines, then proceed to develop the site. The client can see it the whole time (as can anyone else that knows the address). When it is ready, remove the original site, and follow the easy steps at https://www.ads-software.com/support/article/giving-wordpress-its-own-directory/ (which is linked from the Settings > General admin page). Be sure to save Permalinks, and then any internal links in the content will need a slight change to remove the folder name from the path. I usually use the Search Regex plugin for this, to get href but not src.

Edit: if it’s just the theme you are changing, it’s much easier to just use a localhost installation. You can export the theme options and the widget options, and import them into the old site.