[franken]

The gen light is on when the engine is off if I understand things so as soon as the engine starts, the choke would open. I think/thought the electric choke had a bimetalic coil inside that actuates a switch based on temperature, so a switched (on w/ key in run/start) power source would be the right solution and since current is low, no relay is needed.

If you have other wiring problems, I'd leave the choke alone since its more of a luxury item.