What is the proper action when peers in your field have/are publishing papers with a crucial flaw?
This may seem like a cliched answer, but I believe good communication would be key. First off, you need to make absolute sure about whether or not their work represents flaws (preferably with your supervisor).  The next step I believe you should take is to contact the Professor whose team you believe has made such critical mistakes.  Discuss the issue with them and get their perspective.  If, after all of these efforts, you are still absolutely sure that there is a mistake in their work, politely inform them that you disagree with their approach/methods/technique/etc. and that you will be publishing a paper representing a different point of view and approach that presents your argument on the matter.  Mind you, don't ask their permission, it's merely a courtesy so as to show that you don't mean anything personal by it.  After that, publish away.

Finally, I should mention that I feel that there isn't really any issue at all.  If someone got mad at me for publishing a paper that disagreed with their research...that's not exactly a person I would care if they were mad at me or not.  It not only seems excessively petty, but it's also the nature of scientific research.  Great ideas and solid theories get overturned and refuted on a regular basis.  If your counterpoint to their research is solid enough to be published in a good journal, then it's good enough for them to read and consider without getting unnecessarily angry.