Attach the black lead to a chassis negative, red to the positive. Make sure it's a chassis battery and not a house one. With the shore power disconnected and genset off, write down the voltage. Then start the genset, wait a couple minutes, and note the voltage again. Stop the genset and plug in to shore power. Wait a few minutes, then note the voltage once more.
This information should be helpful to others here more familiar with your coach, so they can help track down the problem. Further meter checks will likely be necessary.
It could be as simple as loose battery connections, loose ground strap, or corrosion at the battery posts causing high resistance.
Joel