Ouch. I just can't even begin to imagine how frustrating and uncomfortable this has been for you.
Try the cabbage leaves. There's something in the cabbage that gets absorbed through the skin and relieves engorgement and also works to reduce milk production. Use two inner leaves, rinsed well to remove any bacteria (as the baby may pick something up that stays on your skin), crush the leaves lightly in your hand, and put one inside each bra cup. The first night, try replacing the leaves each time you wake up hurting. Cold from the fridge may feel especially soothing. Perhaps it will help.
I've read that our bodies are especially sensitive to prolactin (the milk-making hormone released by nursing/pumping) during the middle of the night. If I were a lactation consultant looking at your case, I would be very curious to learn more about variations in this whole hormonal picture. It truly is odd that your baby has no symptoms of oversupply (green poops and fussiness, for example) but you clearly are overproducing at night, despite many weeks now of little to no stimulation and milk removal through the night. With your body being so resistent to the change in nighttime nursing patterns, this suggests to me that you've either got lots more prolactin than is typical, or you are much more sensitive to that hormonal process than the average woman. I don't know how someone might go about measuring or determining this, much less altering it, but it's the best theory I can come up with.
Do follow up on your doula's recommendation, try the cabbage leaves, try the sage tea (only at night, I would think), and let us know how you're doing.