Mac HD - Data, MacOS Catalina, and Time Machine

Don't reinstall macOS because that won't help in your case. You need a bigger TM backup device. 500 GB is a paltry amount these days.

As of Catalina, Time Machine no longer backs up an autonomous copy of macOS itself. TM creates a bootable Recovery partition, but the only thing you can do with it is boot Recovery so that you can reinstall macOS.

Forget the old school way of doing things. You can still create archival copies of data in the manner you describe, but TM is much, much more than that. From a completely technically accurate perspective, the required minimum capacity of the backup disk is a function of the amount of data that needs to be backed up. As such, the capacity of the source volume(s) isn't a factor. However, the amount of data that needs to be backed up is not easily calculated. Then, bear in mind TM always retains a minimum of one completely restorable backup even before it begins to create the next incremental backup. Only after it finishes that new backup will it begin to erase older, "expired" backups. When determining how much space it needs, TM always overestimates it. That way, you are never without at least one complete backup.

The bottom line is that if TM says it needs a larger backup disk, then that's what you need.

I wouldn't bother using an SSD for TM. Speed isn't important. Redundancy is.

I have found that TM wants to be at least the size of the internal drive (or larger) or it may state "not enough room" regardless what the actual data size is. That is the way TM appears to want to operate. Can you get another drive (perhaps inexpensive mechanical) that is larger and see if TM will function correctly? I would not reformat your internal drive at this point. You will need to format the new mechanical drive of course.