The erratic idle could be early signs of a bhg, I had the same issue. It was indeed a bhg, but mine was more severely from ignoring the signs. The vacuum and coolant seals on the hg began to seep and then blew. Ofc I had no coolant in my oil or even smelt it in my exhaust but occasional smoke after idling followed by erratic idle when the motor is warmed up. To the point where the vacuum plugged all flow of coolant through the motor, so I replaced the thermostat, coolant temp sensor, and radiator to find out i was a bhg.
I would try to cleaning the fuel system but Im afraid it might be a bhg.
I would try to cleaning the fuel system but Im afraid it might be a bhg.