I honestly don't know if my gen charges the battery when it's running. One would ass-ume so.
But I don't care, I put my smart charger on it at least three times a year and make sure it's fully charged, desulphated, etc.
Smart chargers are the best; that little black box can't have one quarter the brains my Nuco charger has.
Older lady friend bought a dual fuel gen like mine, I think a Wen, on my recommendation and my house present next time is a
little 1amp Nuco genius charger. She can't hand start that big engine, no way. That battery has to work.