Try Pachysandra. I couldn't get rid of Bishops weed in an area until I planted Pachysandra - choked them out completely. aka Japanese Spurge.

Give a couple of years, water well until established, likes acidic fertilizer same as evergreens, and evergreen in the winter. They spread by underground runners like Lily of the Valley, but NOT invasive, easily contained with a border and can be pulled out without much effort. Roots are fairly shallow. My GUESS is pachysandra will win. Within three years with proper care you will have a full bed of pachy. Beautiful plant, in shade and sun - for me they worked just as well provided they received enough moisture when planted in full sun.

Good luck!

Not what you wanted to hear, but contact your local florists...my sister works in the wholesale flower market and told me that June brides often want lily of the valley and it's very expensive...you might be able to make a mint off your lily of the valley!