As suggested above, I would ask your LIA for more info as to why they don't feel a Residence application would be successful. As an LIA myself, I think it's only fair that a full explanation be given as to why, in my professional opinion, I don't believe an application would stand a good chance of success - you deserve that explanation, to guide you in your other possible options in the future.
A drink driving conviction would require a character waiver to be applied for, but this does not necessarily mean that an application is guaranteed not to be successful. The character waiver process is quite straightforward and a good case could be made for a relatively minor incident of driving having consumed excessive alcohol.