Lettuce for Chinchillas?

Hi! I am from Colorado. I have a 14 year old chinchilla who I got about 3 years ago from a rescue organization. I am really worried about her right now because she has pneumonia. We have her on antibiotics once a day, pain meds once a day, and critical care feeding four times a day. Our vet has also recommended the spring mix organic lettuce because her back teeth are too sharp and causing pain. At the moment she is too sick for her teeth to be filed down. The main thing I am wondering is if I should give her the lettuce. Everything that I have read says no to greens and I was surprised when the vet told me to give her the lettuce. My chinchilla has another appointment at the vet tonight. They talked about hospitalizing her but said it would be too much stress for her.
Thanks for taking the time to read this.