$Created is a read-only value. The only way you could change the value would be to edit the TBX document’s source XML outside the app.
If you want the same note in different places in the document, consider using an alias of the original note. Select the note and use menu Edit ▸ Make Alias (⌘+L). The alias is created alongside the original and an then be moved anywhere in the document. Do be aware of the concept of intrinsic attributes in case that matters. for those who think aliases are just a facet of agent use, this shows that most Tinderbox objects (adornments being one exception) can be ‘manually’ aliased.
Otherwise, track a different attribute. What? You say want to track read-only $Created for whatever reason, yet as you note you can’t be sure $Created is always correct for your purposes in all cases.
So, as a new user Date-type attribute $MyCreated. Now, for your main content containers add this $OnAdd:
$MyCreated = $Created;
You can use the same code in a stamp to ‘update’ any pre-existing notes. Now you can also design a method for setting $MyCreated in copy/pasted versions of the notes.
There’s insufficient info to detect further, but it feels like your use if $Created may be building on a false premise. If you’re open to sharing a bit more of the problem, the community here might be able to help. It’s not that using $Created values is wrong but simply that you’ve been caught out by untested assumptions about how $Created behaves (when a note is copied.
FWIW, I do think the current $Created behaviour is correct. It records the date/time that note was created. An alias will use the original note’s creation time. As an alias it is the same note in a different place. But a complete copy of a note when pasted is a new note. It simply wouldn’t make sense to record the creation time of a different note in $Created as they weren’t created at the same time even if the two notes share many other attribute values.
In summary, aliases or employing a user attribute offer you at least two different alternate approaches to avoid the false assumption you’ve tripped on.
HTH, and if not, do ask.