Hi Shane,
OK, this link explains how to replace that SSD:
http://www.everymac.com/systems/app...-to-upgrade-mac-pro-cylinder-ssd-storage.html
However, note that it specifically states that "the SSD uses a custom Apple design with a PCIe 2.0 x4 interface, which is capable of a theoretical maximum speed of 2 GB/s". So, your choice of third party SSDs that you can use seems to be very, very limited to ones sold by OWC. And the 1TB SSD from OWC that you can use costs $880. I really do not know if any other companies sell that "particular" kind of SSD.
Now, regarding your stand alone SSD question, yes, it is definitely possible to connect an SSD externally to your Mac Pro and run the OS from there. Basically, you can purchase, for example, a 1 TB Samsung 850 EVO SSD, install it inside a nice, slim Orico external enclosure with USB 3.0 connectivity, connect it to your Mac Pro, Erase and Format (and possibly partition; that would be up to you) the SSD with Disk Utility, and then use SuperDuper! to backup/clone your system to that SSD. You can then select that SSD (or the partition containing the SuperDuper! backup) to boot your Mac Pro.
I recently purchased 2 Samsung 850 Pro 512 gig SSDs (got a very good price), installed one of them inside an Orico 2189U3 external enclosure (
https://www.amazon.com/ORICO-Enclosure-Super-Speeed-External-SATA3-0/dp/B01LXY78V6), and did exactly what I stated above (I have 3 partitions on the SSD; one for each of my Macs, and the other containing miscellaneous "stuff"). I make SuperDuper! backups for each of my Macs to their respective partition once a week, and I can boot either machine for its partition. The USB 3.0 connectivity is "decently fast". You can purchase external enclosures with Thunderbolt (maybe Thunderbolt 2?) connectivity, but they are rather expensive. The Orico one for me is fine.
I need to mention that SuperDuper! does not backup the (hidden) Recovery HD partition. Carbon Copy Cloner does, and like SuperDuper!, it makes a bootable backup. For me, though, that is not an issue, as 1) I have more robust disk maintenance/repair software than Disk Utility, and 2) that partition gets created with a fresh installation of the OS. SuperDuper! costs $27.95 (there is a demo version you can use), and Carbon Copy Cloner costs $39.95:
SuperDuper! -
http://www.shirt-pocket.com/SuperDuper/SuperDuperDescription.html
Carbon Copy Cloner -
https://bombich.com/
However, in your situation, given that the 1 TB SSD went "bad", I'm not sure if you have such a "source". If (hopefully) you've been making backups to an external device, that is a possibility. But, if it's a Time Machine Backup, not sure how all of that would work.
So, if you could let us know what the Apple Store is specifically doing, and about backups you might have, that could help.