That said, I think there are specific contexts wherein a euphemism becomes a barbed joke. Say a bunch of horrible people are thinking some runway model is fat, and a photographer says the model is 'curvy', and people laugh. The insult has been veiled with humor, and the humor is the veiling itself. That is, it is funny because an insult was lanced without being made explicit. This is a very particular case of overlap with the case I was describing. But again, exploring the areas where 'euphemism' is a correct description reinforces my feeling that it hardly overlaps with barbed joking.

You are proving my point for me. Curvy for fat is a euphemism, but it does not contain either humor or an insult. The only way these things could be attributed to 'curvy' is through context. In this case, it would be a barbed joke expressed as a euphemism, and the context does all the work.

No, this is not right at all. Thanks for your contribution, but a euphemism is far more general than the concept in question, and hardly even overlaps with it. An example of a euphemism is 'passed away' for 'died'. In this example there is no insult, and no humor, so both of the required concepts are lacking.

Ha. That's funny!

I'm here, none of them are even close, though I appreciate yours for its thoroughness. I've been trying to find the word in my own browser history and with other searching without luck. @Catija has had the best idea so far.