11 months and tanking supply

MLESSER100

My LO is 11 months old and has been EBF. Over the past month my supply has been dropping to nearly nothing. After 2 minutes of nursing DS becomes frustrated and stops eating. I pump before I go to sleep every night and the past few nights I've only pumped 1 oz. 

Now here's the dilemma: We still have a month until he turns 1 and we can introduce cow's milk, so I can't wean yet and every time we have tried to give him formula in the past, he wouldn't touch it, so formula isn't really an option either.

Any advice would be appreciated. Thanks!