Personally I'd cut it, my dad always said Sept.5 was the last day to cut hay until after it starts freezing at night and I'm in E.C. WI.
I tried direct seeding alfalfa this year (April 12) for the first time and after each cutting it seems to get better. I spread 250 lbs. potash last week and if we receive more rain I may need to take a fourth cut.
Pic is of the second cutting (July 11) and was kinda thin because of lack of moisture but I felt the need to at least manage the weeds in case it did rain.