Her low temperature is almost assuredly causing her blue lips - low body temperature is a major cause of blue lips, and her temperature is LOW.
What's more important here is to find out why her temperature while sleeping drops so low. Does your doc know her temperature is that low? That is absolutely something to look into and diagnose.
A google search, for what it's worth, revealed that possible reasons could be hypothyroidism and diabetes issues.
I was gonna suggest thyroid tests as well....that's scary. If your doctor can't help you, you need to ask to be referred to a specialist who can find something definitive...I know you're scared, just make sure you tell your doctor that you want an answer no matter how far you have to go for it.