Jeff,
Have you considered using some Arrest on the Alfalfa? It will kill the weeds and not bother the Alfalfa. For that matter, you can use Arrest on your Clover too- but I do agree on cutting the Clover to help choke out the weeds as well.

If you cut the Alfalfa, then you get hit with a week long heat wave- it may stunt the growth somewhat and/or really dry out your soil as now you have more sun/heat reaching the soil rather than being absorbed bythe alfalfa. How low were you planning on cutting the Alfalfa downif you cut it?

I say, go with the Arrest. It's a selective Herbacide that'll get rid of the weeds you seek to lose.

Rob