M.R.
ADDED: Saw your change to the post and am glad the doctor knows what you're trying. I would just phone their office, say she told you to try coconut milk, and ask the advice nurse how to proceed. I agree with Mynewnickname above that you might want to wait until your child is no longer coughing. You won't be able to tell if the coughs are what she's already dealing with, or something new caused by the new food. Good luck!
ORIGINAL: I know you've posted a lot lately with health concerns regarding your child. I really would not introduce potential allergens, if that's another concern, based on what laypeople say; I'd consult first with your pediatrician or even get an appointment with an allergist who specializes in working with young children. You will get a lot of different opinions from different web sites and different people posting. But this is really an issue for a professional to handle, especially as you've been very concerned and stressed lately by your child's health; you may be least stressed if you just go to a professional who can lay out steps for you, rather than asking parents--our methods and experiences may be totally different from what your child needs.