Ok; salt, and then hash your bucket names
If you mean to use a "secret" prefix (i.e. pepper) then, that would generate effectively globally unique names each time (and unpredictable too) but you can't change the pepper and it's only a matter of time it'd leak.
The public/private distinction seems moot here, too: the salt is a throwaway since you just need the bucket name.
Even if you do need to keep track of the salt, it should be safe for the attacker to know, at least with respect to this attack, because you already own the bucket which the attacker would otherwise hoard.
1. You set up an aws bucket with some name (any name whatsoever).
2. You have code that reads and/or writes data to the bucket.
3. You delete the bucket at some later date, but miss some script/process somewhere that is still attempting to use the bucket. For the time being, that process lies around, silently failing to access the bucket.
4. The bucket name is recycled and someone else makes a bucket with the same name. Perhaps it's an accident, or perhaps it's because by some means an attacker became aware of the bucket name, discovers that the name is available, and decided to "squat" the name.
5. That overlooked script or service is happy to see the bucket it's been trying to access all this time is available again.
You now have something potentially writing out private data, or potentially reading data and performing actions as a result, that is talking to attacker-owned infrastructure.