At the moment, users have been logging on with 'CURRICULUM\[Username]'. However, we have recently started using an app that refuses to change the domain name from 'corp' (we are currently in the process of resolving this issue with the software company).

In the meantime, is there a way users can also use the 'corp\[Username]' to logon? I have tried adding the OptionalNames in the registry of the DC, however, this doesn't seem to work when logging in on a client.

You cannot change this name unless you are renaming the entire domain. And this is a very risky operation because it impacts all users and resources, and many applications just don't support domain rename, see here: How Domain Rename Works.

Maybe your application can use the UPN format as a login format. it is the one that looks like an email address: user@contoso.com. If so, you can add additional UPN suffix to your forest and you can change the UPN of accounts to something like user@mynewupn.com, see here: Add User Principal Name Suffixes.