Data should be backed up to staging nightly anyway. There should also be scripts in place to start this process at an arbitrary point in time and to import the data into the staging server. You do not need to match the hardware if you use cloud hosting since you can scale up whenever you want.

Here's where it gets really simple. Resize the staging instance to match live. Put live into maintenance mode and begin the data transfer to staging (with a lot of cloud providers, step #1 and #2 can be done in parallel). As soon as it finishes copying, take live down, point the DNS records at staging and wait for a few minutes. Staging is now live, with all of live's data. Problem solved. Total downtime: hardly anything compared to not being prepared. Total dataloss: none.