I'm a big fan of peppering. It strikes me as a defense in depth move. Say a programmer on your team accidentally pushes code that makes your salt a static value for all users going forward, the pepper might protect user passwords in that case.
This should fail an obvious unit test for the password handling code. The obvious unit tests for this component include a check that minting two password hashes for the same password gives different results.
If people are going to be allowed to meddle with security critical code with no effective oversight you're already dead you just didn't know it yet. They can go modify the password checking code to return true because that seems to "work better" and if nobody rejects that you no longer have password authentication.
I don't see any value of a pepper. It doesn't actually add any protection.
Modern password hashes like argon2 include this feature natively.