If you have created 'email aliases' (additional addresses) you can't make one of those into an Apple ID.
You should be able to sign in with the main @icloud.com address (the one with the same text before the '@' as the @mac.com address) anyway (it's effectively the same as the @mac.com version), but also you should be able to change to that at https://appleid.com
You say you 'can't get emails' on the @mac.com address: if you are getting emails on the @icloud.com version you should also get them to the @mac.com address. There should be no problem using the @mac.com address as your ID - this dates from the days of DotMac and the @me.com and@mac.com versions were added as MobileMe and then iCloud came along as replacements.
If you continue to be unable to either receive emails to @mac.com or change the ID to the @icloud.com version you should contact Support: https://getsupport.apple.com