Enable NTFS

That is the answer to your initial question. Stop trying to use NTFS on your Mac. There's no need to use a format that only works perfectly with Windows when ExFAT works perfectly with both Windows and OS X, and requires no third party software to read or write to the drive. If you have no need to use the external drive(s) with Windows at all, then format them as OS X's native Mac OS Extended.

Also, you say you've tried numerous NTFS add-ons, but did you uninstall each one before trying the next? Having multiples will really goof things up as multiple active drivers will all be trying to do the same work, which will cause them to trip over each other.

If the drive is only intended to be used with a Mac, then format it for a Mac, which is Mac OS Extended.

If the drive is meant to be shared between a Mac and Windows, then format it on the Windows computer as ExFAT. This format is read/write on both platforms without the need for any third party software.

You will need to move the data and reformat to ensure that there are no issues in the future. You need to format to Mac OS Extended Journaled in GUID partition table to make sure there are no future issues.

2 TB of data on one drive isn't a backup. As soon as the drive fails (and ALL drives eventually fail), it will all be gone. Invest in a second drive and copy the data to the newly formatted drive. As mentioned earlier, use Mac OS Extended if you will only ever need to access the drive from your Mac, or as ExFAT if moving it between a Mac and a Windows computer.

Once everything has been copied to the new drive, you can reformat the original to match the new drive's format and copy everything back to it. Then you will actually have a backup.

Actually, since you're already topping out the drive you have, I would buy two 3 TB drives to clone the current drive to twice. You're going to need more room from the sounds of things, so getting a second 2 TB will just leave you with two drives with no space left. The 2 TB drive you have can then be reformatted for the Mac and used to backup the Mac's internal boot drive so you have another way to start up your Mac when the internal drive fails, or needs repair.

Why do you think you need to keep NTFS formatted disk drives on a Mac?

Having NTFS on a Mac is a good way to cause problems for yourself as you have started to discover. Really the best solution is to get rid of NTFS on a Mac rather then using it.

You may not want to reformat it but it is only going to cause you problems in the long run. You might as well do it now rather then waitting until later when you discover the next incapability between Mac and NTFS because you will.

Another disk drive with at least 2 TB of storage. Simple.

The ability to write to NTFS is unsupported by Apple and is not recommend to be used.