Meat should pass through your system without needing fibre. Your body should be be digesting so much of it that there is very little left to pass out the other end. The meat should be fairly fatty and not over cooked though.
Celiac disease symptoms are caused by eating foods containing gluten. If you are not eating those then celiac disease shouldn't be a problem.
Are you eating nuts? I really like them but they cause me problems. What are the typical foods you eat?