My thought would be to go ahead with the lasagna bed and not worry about the fairy ring fungus. I think fairy ring fungus will neither negatively affect your lasagna bed nor the flowers therein. It's a grass thing.
Bill Hill

Fairy Ring occurs most often in turfs with excess thatch, although it can be seen in soils with very high levels of organic matter. Other than being a bit unsightly it is not a terribly long term problem since the grass grows back pretty much by itself as the mycelum move on. Most often if I see Fairy Ring it will be in turf grass that has been fed too much synthetic fertilizers, "weed" killers, and pesticides and as a result the Soil Food Web is virtually non existent in that soil and thatch builds up to unhealthy levels.
Yuo can build you Lasagna Bed over that with no problem since, as Bill said, "its a grass thing."

Thank y'all very much for the advice :)