"Creeping Charlie" Infestation

"Creeping Charlie" (Glechoma hederacea) ground ivy has infested about 40% of my back yard. The trouble is, I raise Embden geese for food in that area which eat the grass within. I won't use any pesticides because I don't want cancerous chemicals ending up in my geese which would end up in me. Nor will I use borax which might render my soil ungrowable to plants for all eternity. So how do I kill this miserable pest which contains pulegone, a marginal liver toxin. The geese won't touch it, and from what I read online it is somewhat bad for goats. A few people have used chickens to eat it down to bare soil, but other than that it seems the only real way to obliterate it is to re-sod the entire lawn. A buddy suggested dumping boiling water on it, but on 1/4 acre? If nobody has a better idea I'll section off the area and spray muriatic acid on it. No carcinogens and the rain will wash it away. Alternate suggestions appreciated...

Try a vinegar based herbicide (might need a couple applications, it only burns foliage) or cover it up for weeks to smother it out?

They do make herbicides specifically for ground ivy, but I can't say much about safety and toxicity.

I agree with the covering it up idea. If it were me I would try a couple of approaches to see what works best.

Worth thinking about is: what are you going to plant when it is gone?