isn't necessarily too
slow; it's going to copy at full HDD read-speed, it's just that it'll as said also copy file system blocks that are not in fact in use. With 1T and, say, 100M/s HDD read-speed that makes for 1000000M / 100M/s = 10000s ~= 2u45m. At 50M/s average for your HDD 5u30m. In reality it's likely somewhere in between those.
If however as said there's not a huge amount of free space anyway that number also doesn't change much -- and it's always going to be long. Long vs. Very Long tends to not be as big of a difference as Short vs. Long. I.e., you'd be doing something else in the meantime anyway
When a few years ago HDD -> SSD replacements were very common external cases tended to be supplied with
(Windows) cloning software. Maybe still, not sure, neither sure said software would be able to skip unused blocks on ext4. Something like Acronis True Image for Windows probably does and for Linux there's Clonezilla and also the recently by a Linux Mint forum-user (AndyMH) developed Foxclone: https://foxclone.com/
As to names: no, you wouldn't be able to change names like "sda" and "sdb"; they are kernel-assigned and depend only on discovery order: if your SSD will be the only SATA device then it will be "sda" in the same manner as now your HDD is. If you'd install your HDD in the case you'd probably repartition and reformat it anyway, so in that case no issue either; only when you'd connect it as is would you want to change UUIDs on the HDD. This is easy to do once you have things set up.