I think the big issue is that joking about school shootings is satirical to bring up how common it is in hopes to change it, while joking about fat people is just to be mean. No one is going to change because you made fun of them, in fact it might make the problem worse as state of mind is a factor in being unhealthy.

You could probably make the same argument about abortion jokes being satirical, but I don't think abortion jokes are made in order to enact social change. It's usually just to be offensive. Satire is usually used to point out how ridiculous something is, the hope is that we as a society will see it and do something about it.