I'm not sure why the court would overturn a verdict of guilty of causing emotional distress because of free speech laws. Free speech isn't (or at least shouldn't be) license to do something that you purposefully know could torment someone.
Free speech should definitely include the right to say things in public that torment people. If free speech didn't include the right to, say, protest outside the white house chanting "Hey hey LBJ, how many kids did you kill today," despite the emotional distress it caused Lady Bird Johnson, I don't think it would mean much. Or say you want to protest outside the Westboro Baptist Church, with signs that say God loves everyone. That message, you know full well, is very upsetting to the Westboro Baptists. Should you have to risk paying them money for their emotional distress? I think the answer should be no.
So as a general rule, I think free speech should protect protesters from this kind of liability. But there are two reasons, taken together, why I think the protest at a funeral should not be protected. First, the family of the marine did nothing to put themselves in the public sphere, and they were attending an event that, though in a public place, is private in nature. Second, the signs were directed specifically at them, criticizing their behavior personally. Free speech should protect unpopular political opinions, no matter how repugnant, to the ends of the earth, but it need not extend to protecting what amounts to an attack specifically directed at a private individual in a private place.