UPDATE: The job is now under my belt.
The first thing I did was to have a sit-down talk with the drummer before the show. I asked him what instrument he wanted to be the loudest in the mix, what next, and what should stand out the least. He was actually helpful and accurate in describing his goal and it gave me something to go on.
He wanted a "dance club" type mix with heavy low end. He told me that the backing track included lead vocals because his singer really wasn't that good. So, I needed to keep them equal. He also explained that the guitar player was merely lip-syncing his guitar with the keyboard on the backing track, that I needed to keep his level very low except during real guitar leads.
Now that I finally understood, I was able to work with them (not against them) as I mixed their show. I am happy to say it went pretty well. Someone must have told him that it sounded good because he (the drummer) gave me compliments at the end of the show.