My plan would be to go ahead and add the new NVME as a pool, name it cache if you wish, doesn't matter much as long as you keep track of the name, then disable docker service (not just the containers) and VM service if the VM image is on disk6, then set up the mover to transfer all the shares currently on disk6 to the new pool. That way you don't have to do the backup and restore.
All this hinges on being able to have both NVME drives installed at the same time, if that's not possible then your only safe option is to copy the files elsewhere off of disk6 like you were planning. If you are ABSOLUTELY SURE that all the rest of your disks are 100% healthy, you could always remove the current NVME and allow the rest of the disks with parity to emulate the files that were on disk6 and copy them to the new NVME assigned to a pool.
Nothing "special", but there are so many ways of accomplishing what you want to do, it's tough to pick one way as best. The only thing I would warn is to make sure the docker and VM tab aren't present in the GUI when you start copying files around, that way you can be sure that there won't be any in use system files that don't get copied. Disabling the services will remove the tabs from the GUI until they are enabled again in settings.
At some point in the process you will have to follow the shrink array procedure as mentioned by JorgeB to remove a data disk slot, unless you replace and rebuild it with another 4TB. Given that you have 10TB free you really don't need another 4TB of space right now.
Sorry if I confused the issue with more options, be sure to ask more questions and lay out your plan with more specifics before you actually do anything, that way we can look over your steps and verify whether it sounds safe.