Liza at the Palace is a revue/one-woman show, not a musical. Will Rogers Follies absolutely does not fit this definition.
Pump Boys and Dinettes and Oil City Symphony are both pretty close to what you are describing. I'm Getting My Act Together and Taking It On The Road is staged as a rehearsal for a concert/nightclub performance.
"You travel alone because other people are only there to remind you how much that hook hurts that we all bit down on. Wait for that one day we can bite free and get back out there in space where we belong, sail back over water, over skies, into space, the hook finally out of our mouths and we wander back out there in space spawning to other planets never to return hurrah to earth and we'll look back and can't even see these lives here anymore. Only the taste of blood to remind us we ever existed. The earth is small. We're gone. We're dead. We're safe."
-John Guare, Landscape of the Body