While it may be expensive, fertilizer is the way to eradicate them. They can't take the Nitrogen. It really won't take that long. You don't have to use a balanced fertilizer, just a high concentration of N. (ammonium sulphate, urea, etc) Even better, if you have access to chicken fertilize, that'll do it and it's cheaper.
How much nitrogen? Could you overseed clover to increase nitrogen and then just kill off the clover with 2-4D? Or does it take a heavier dose?