Will the bile acid in my bowel movements irritate my rectum?

Yes. It is not the bile acid - it is the acidity of the food that creates this issue - food becomes acidic if it is high in animal protein people who do not digest this has higher Lactic Acid formation, food rich in trans fat (fried foods, oils) and food rich in sugar burns the mucosa of gut where there are toxic focal sites - they loose intestinal lining and this burns when it comes across bile acids.
Should not. Bile acids are normally secreted into the small intestines and are a normal part of the digestive process.