Let's regroup and start over. Your File Manager should perform this task without the risk we are creating by using sudo on the command line. If Linux Mint or your File Manager is broken, then let's try to fix that so they work properly. Copying files is a basic task that needs to always work without issues.
Your previous Ubuntu developed problems, so you have installed a fresh copy of Ubuntu (that contains the files you want) and you installed a fresh copy of Linux Mint (where you want the files to go). You also have your backup files on USB... but you say the USB is "write protected" and won't copy. This is all correct?
I use Linux Mint myself, so I can follow along pretty well... but I am not familiar with LVM partitioning. If your File Manager is working properly, though, I don't think that LVM makes any difference and should not complicate this problem. Someone please correct me if I'm wrong.
So, guide me though this on your system, as it compares to my system (don't plug in your USB yet)....
1. I run Linux Mint and open the File Manager (well, Mint calls it a "File Browser"... located in the "System Tools" category, or the "All" category, or click on the panel icon next to the terminal).
2. The File Browser opens up in my home folder /home/stan. Clicking the UP arrow on the menu bar goes up one level to home (shows the stan folder, but I have no other users). Clicking the UP arrow one more time takes me to the root of my system ( / ).
3. From the root ( / ), I double-click on the media folder, where I find another stan folder. I double-click on the stan folder and go inside... it is empty. If you've jumped ahead of me, you may see your BACKUP DATA folder here already.
4. I insert a USB flash drive, and it appears. Mine has no name but just a kind of serial number.... but if the USB stick has a "label" assigned to it, that should show up (so yours may show BACKUP DATA). My folder also shows a small "X" in kind of a superscript to the USB drive's folder icon. When first inserted, the USB stick is controlled by root so this "X" indicates limited permissions.... but that is not really true.
5. Even with the "X" still showing, I can right-click on that USB folder icon, with is serial number ID, and I can choose Copy. I can then navigate with the File Browser to another location, and I can Paste that entire folder with the serial number staying intact. I pasted into my Downloads folder just to test. Also, while still in /media/stan and looking at the folder with the "X" still showing... I can double-click that folder to look inside, and then go Back (or use the UP arrow again) and it will show the folder and the ""X" will now be gone.... and the permissions are now set to me, instead of root.
OK, sorry to be long and tedious.... but if your system does not perform like this, then I hope to try to understand better what is failing you. Most USB sticks do not have any kind of "write protect" available, but even if they do, it should still allow a Copy operation (that is a read operation, not a write).
If you follow the latter instruction in #5 above, and open your BACKUP DATA folder, and then UP arrow out of it... if it still shows the "X" superscript, then you probably have other permissions applied to the USB stick which you need to reduce so it will allow the copy.
Hope that helps!
Cheers