I de-wild garliced my lawn by using a bulb planting tool. I was able to lift the core out, peal off the top sod, and pitch the dirt ball containing the bulbs and baby bulbs. After replacing the removed dirt... You can just slap the sod back on top. In your case, I would think you could do the same (except not have to worry about saving the sod part)....you could work around most of the ivy runners without causing too much damage... That wouldn't just fill back in after a short time. Isn't it hard to kill ivy even if you wanted too?

Now if you have clusters of this stuff every 6 inches... That's a different story.

I have clusters ever 3 inches - this is an invasion.

Onion soup. Lots and lots of onion soup.