Rabbit starvation only happens when 2 things occur
- there is a LOT of lean a lot of protein in the diet, and
- the diet does not contain fat
Just like you were eating rabbit every day. https://en.m.wikipedia.org/wiki/Protein_poisoning
So any diet where protein is not eaten to excess, and fat is eaten, will not cause rabbit starvation.
Foods such as pork, beef and lamb, chicken (with skin), salmon and other oily fish all naturally contain enough fat to avoid rabbit starvation without needing to add further fat.