Sometimes the mayor is just stubborn about taking a drink. It might help if you remove all other types of alcohol from your inventory other than Rimmer's ale.
However, giving the mayor a drink will just allow you to get a password from him, which you do not really need. Simply head to the mine in the dwarven halls; using the password would get you past a sentry there, but it is an easy fight that gives more experience than the peaceful solution (you can not use the password and fight him later).
