When an organic grower compromises and uses products not acceptable , such as any synthetically produced herbicide, they are no longer organic growers. Keep in mind that the manufacturers of many of these products have fudged the data and it can take many years before the true problems with them appear.

I'm a practitioner of "Integrated Pest Management" - whatever works. I would get rid of the Creeping Charlie by the least invasive method possible, then work on a routine that will keep it out.

It likes moist areas ... make sure you have good drainage and aren't overwatering. Use a moisture meter and adjust your watering based on need instead of calendar.

Make sure you aren't mowing too short, and fertilize as needed to get a good dense turf. Use the right kind of grass for the conditions.

If it's really well-established, and it sounds like it is, use whatever herbicide that is most likely to kill it, exactly as specified on the label. Then carefully remove the corpses and reseed the bare areas.

In the future, work on keeping the lawn conditions so they aren't as hospitable to the weed. Check for sprouts weekly and pull them out.

No matter how you attempt to justify using synthetic materials, if you do you are no longer an organic grower. Just a little, just once, does damage to the environment just as continuing use will.