there's no way to avoid offending them while also acting rationally. It becomes necessary to choose.

Really? Have you tried, maybe, just not pointing out the weaknesses in their arguments? Mightn't that be the rational thing to do? Just a polite smile and nod, or a gentle, "Have you considered some alternative?" Or even, "You may well be right." (This is true of pretty much any non-contradictory statement.) Or there are many different ways to argue with someone without being confrontational. Asking curious-sounding questions works fairly well.

It's generally easy to recognize how well a person will react to an argument against him. If you have basic people skills, you'll be able to understand what type of argument/approach will communicate your point effectively, and when you simply don't have a chance. The idea that it's necessary to offend people to act rationally seems completely absurd (at least in this context). If it's going to offend them, it's going to accomplish the opposite of your goal, so, rationally, you shouldn't do it.

This whole discussion reminds me of the Dave Barry quote that may well have been used earlier on this site:

"I argue very well. Ask any of my remaining friends. I can win an argument on any topic, against any opponent. People know this, and steer clear of me at parties. Often, as a sign of their great respect, they don't even invite me."

This. Is. Not. Winning.