I would first go to the site history and mark the last published version as a " back up" , to switch back to this version if things don’t go well as planned while editing.
Then I would start redesigning. As long as i create a release candidate, instead of publishing, the changes will not be carried over to their live version(last published version), hence they can still update and keep using the live site
Use the R C version to test or share for reviews. Once reviewed, you can publish the RC version. You can only create one candidate at a time; will have to delete old one before creating new. One more thing, even if you don’t publish changes and only create RC, the client will still see the new changes you make when they open editor. If they only need to update stuffs to the live site without opening editor, then this might work for you.
All the best!