Would a rum flavored cigar cause one to fail breathalyzer?

No.

Breathalyzers measure the blood alcohol content (BAC) in a person's breath. Alcohol is absorbed into the bloodstream through the stomach and small intestine. Smoking a cigar, even one that is flavored with rum, does not introduce alcohol into the bloodstream. Therefore, smoking a rum flavored cigar will not cause a person to fail a breathalyzer test.