I've been in this situation, one mic having to do both duties, with no easy access to the board to alter volume levels and no sound man. I simply turned up the volume of the mic, to adequately offer enough volume for both my instrument and voice and then alternated the distance that I approached the mic with either. I also altered my attack on my instrument.

This is a solution that many bluegrass bands use in these parts to feature both soloists and vocalists. I've seen a whole band using just one centrally placed mike. However, that is acoustic bluegrass music. The band in question may not be acoustic in nature, especially if they are having trouble hearing the banjo. Most banjos, that I'm familiar with, have no trouble being heard but I know that there are different styles of banjo and different approaches to playing them.