Can I clone my Time Machine backups folder to an image file, to essentially backup the entire Time Machine?
The answer is yes and the tool I would use for this task is Disk Utility.

You can use the File menu to select New Disk Image from Folder… and choose only the Backups.backupdb folder as the source of the image.



You can start and see how it works, but I wanted to add some commentary as well since you already are seeing difficulties with other utilities explicitly not wanting any part of cloning Time Machine data.

Since this folder contains hard links, if you don't choose a tool that respects hard links, you will end up with individual copies of each file and not the efficient storage that Time Machine uses.

For network use, you might choose a sparse disk image format which will be easier to copy to S3 and not lose the entire file if the transfer aborts in the middle.

If I were you, I'd perhaps pick important backup intervals and only move those to the offsite storage, but networks are fast enough that you could push an entire backup set to the cloud if you didn't mind paying the data transfer costs and waiting for the upload to be written to S3.