Success! I've found some modern 2TB disks that seem to work perfectly fine with the DNS-343: The Western Digital Red NAS WD20EFRX drive. You can find this drive currently in 1, 2, 3 and 4 TB versions. The 2TB version can be found for about $100 currently.
I've plugged 4 of these into my DNS-343 and it initialized RAID-5, formatted EXT2 with no errors and SMART shows Pass for all 4 drives. I'm currently copying over my data, all seems good.