She is too young, period, cold or no cold, to be left to cry. This is the issue with cry it out for infants: It flies in the face of what we know about babies' brains and their attachment to their adults.
Please understand: At seven months, she has zero concept at all -- none -- that you still exist when you are not in the room. Infants have no idea that people are there in another room; when you leave her sight, you are gone forever, never to return, and she is not capable of understanding or learning to understand (for a long time yet) that you are "right here next door." When you leave her to cry it out, she is not learning to "self-soothe" at this young an age; she is learning only that when she is distressed, no one comes. This does not teach self-soothing that ends in baby sleeping; baby just exhausts herself crying and then falls asleep. It does not teach her to be independent because infants are not designed to learn that lesson yet; toddlers, yes, but not infants. Infants are pure need: I need my adult, my adult does not come, my adult is not there or anywhere else--just Is Not. Ask your pediatrician if you want -- they can tell you that at seven months she cannot understand that you're "there" if you are not where she sees you.
Please stop CIO until she is actually old enough (which could come around age one to 18 months) to grasp that you are actually alive and there when she cannot see you. Right now, and for some time to come, she does not know this and cannot be taught it by letting her cry it out. She is simply not developmentally able to be taught this. I know the CIO fans out there will say all this is untrue but even our pediatrician with three kids herself says CIO goes against what's known about infant development and makes babies less secure, not more secure, and clingier when you are there, rather than more independent. Those are lessons for later. The lesson for her at this young age is that when she expresses a need, someone does come. That makes her more secure, and kids who are more secure will actually be more independent later.
I know you are tired but frankly this is life with a seven-month-old baby who is hungry three times a night. She can't control that or her sleep patterns no matter how much your husband wants her to. What he wants her to "learn" is something she is not able to do right now and he, not she, has to adjust. This WILL pass and fairly soon. Meanwhile, it's time to pump and get her on bottles some of the time so your husband can help feed her.