My answer for something of this nature is it depends. I checked out the airspace around OP's airport and with no way of verifying you have a working charging system and not needing to get home I think I would have done the same.
OTOH I have departed knowing I have a charging system problem. In that case I wasn't in a terribly busy area, was far from home, weather was clear, and I was going directly to an airport where I could get repairs done.
OTOH I have departed knowing I have a charging system problem. In that case I wasn't in a terribly busy area, was far from home, weather was clear, and I was going directly to an airport where I could get repairs done.