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.

I recently read a paper that I think does something quite similar to what you're thinking about. You can access it here: https://physics.nyu.edu/sokal/complex_dynamics_final_clean.pdf but here is a citation for posterity:

Brown, N.J., Sokal, A.D., & Friedman, H.L., (2013). The complex dynamics of wishful thinking: The critical positivity ratio. American Psychologist, 68, 801-813.

This is a case in which a psychological theory was built upon some concepts taken from other fields and the authors here argue they were both wrong to do so and at any rate seem to have made key errors in their applications of those concepts. To be clear, this example is pretty much scorched Earth; the critics are not pulling any punches or trying to make anyone feel better. Of course, you have to remember that anything you say will be read against the literature that came before it, so you need to make a strong case.

What I think is done well here is:

• Make clear what exactly is wrong and what the implications are for the research area.
• Be absolutely certain about the correctness of the criticism and painstakingly explain it so non-experts have a chance to evaluate your new claim. Chances are, the error(s) has gone uncorrected due in part to others not knowing enough to notice it.
• Discussing a potentially correct use of the concept (this could be where your research comes in).

I should also note that the lead author of that paper was a graduate student in psychology who somehow connected with the second author, an eminent scholar in mathematics and physics. It's also clear from the author note that the author(s) reached out to several other highly-qualified researchers to solicit feedback. One lesson here is it is helpful to get powerful allies, who are often going to be people who don't have a dog in the fight (but could be people who always disbelieved the research you're criticizing). Another is that it pays to be really, really sure you're correct.

In this case, the lead researcher behind the criticized theory responded and gave substantial ground to the critics (link to gated article), saying she was not qualified to dispute the critics' claims and felt the gist of the theory was correct but clearly the mathematical side was not. I wouldn't be so optimistic about the original researcher conceding a whole lot in your case, but it's a good example of how an argument can be presented so convincingly that everyone involved finds a measure of agreement.