I don't know why you want to tap into the low beam if your DRL's are on low beam.
Try this circuit with 2 relays and see if it works for you.
Fog lights can be activated with parking/low beam lamps and will be disabled on high beam.
I figured this out years ago
it only happens in the winter months, people can't be bothered to change a headlamp because it's too cold, they wait till spring to change a bulb.