I totally agree with the other comment. Now is not the time for extended fasting. Lots of green tea during the fasting window and a nice hearty bowl of hot chicken noodle soup is what you need! Get well soon!
There are a few recent papers showing positive links between BHB and immunity. A Google search of BHB and immunity should lead you to the papers in the National Library of Medicine and Nature. I hope you manage to dodge the illness.