Choosing between hash functions instead of providing a salt for a password hash
Generally you don't want to switch algorithms in cryptography. You'd have to deal with different workloads. Furthermore, the choice of algorithm would be just another salt, if a tiny one. However, your approach doesn't disallow the building of 5 separate rainbow tables. So the reason for a salt isn't met.

It is unclear why you would not be able to use a general Password Based Key Derivation Function (PBKDF) / password hash. So yes, using a well known good scheme with a a salt is indeed more secure.

A pepper can certainly be helpful; if the DB is stolen minus the pepper then it will be next to impossible for an adversary to start guessing passwords without the pepper. Of course the pepper should then not be kept in the DB, because that would negate the whole idea.