Just to give my two cents here: this is in fact not an easy task. WordPress is not built for these kinds of operations. This is, IMHO, one of the biggest pain points of WordPress (but also other CMSs as well).
This also a classic problem for every web shop owner who does not want to lose products, orders and customers on a redesign.
I really like to use the WP Staging Pro plugin to easily create staging environments.
You can create a staging website copy from within the plugin. Then work on that STAGING site, and push back your changes to LIVE when you are done. You can even create several staging sites or even a staging from a staging site. In fact, this is kind of a classic development setup: LIVE SITE > STAGING SITE > DEV SITE
The hard task is how to not overwrite your existing users. I used two methods in the past:
Omit Certain Database Tables
In WP Staging, when pushing your updates from staging to live site, you can choose which database tables to omit. The question is, which database tables contains the information, you do not want to push back to live site. In this case, MemberMouse devs should be able to tell you, which tables to omit to not delete any new members on the live site.
But you have to test (!), if this is really working. I.e. example create STAGING and immediately create a DEV site from STAGING. Now create some new dummy members on STAGING. Then do some design changes on DEV. Now push DEV to STAGING (omit the database tables that contain the members’ data) and check if none of the dummy members has been deleted on STAGING. If it is working, this might be the setup to push from STAGING to LIVE on relaunch day.
Manual Export/Import
Another way would be on the day of the relaunch export your members from LIVE, put your LIVE site to maintenance, now push STAGING to LIVE. Now import your members you exported before. Depending on the import process, you need to delete all current members before importing to avoid duplicate members. Maybe Member Mouse has some export/import methods? Or maybe Plugins like WP All Import/Export can help.
Just some basic ideas/concepts on how to possibly achieve this. Again, this is not a trivial task. Good luck and Planckspeed