Circumventing the 5th amendment by granting immunity then imposing contempt of court.
Sounds like great grounds for appeal.
E.g. in this case the accusation is that Olson vandalized BoA for 6 months, causing the bank to have to continually pay to have the chalk cleaned off and risking further property damage.
Olson has a free speech right to protest, but that does not mean that he can unilaterally choose means that infringe on the rights of others.
You are right that foreclosing the issue may make great grounds for an appeal though, it's possible an appeal court could rule that the First Amendment is relevant and cause the case to be re-tried.
To not even be able to reference his right to free speech, if it's upheld I'm quite worried.
If his right to free speech is indeed being violated by applying the law, then modern legal practice does not consider this to be an issue for the jury itself to consider (it is considered in a separate manner, by the judge and by appeal).
In both circumstances, participants must follow the speaking rules or be subject to penalties of various sorts (contempt of court, censure, etc).
Attempting to utilize 1st amendment rights in order to bypass those procedures seems like the wrong approach. A more logical avenue might be appeal based on the 6th Amendment right to a fair trial and/or due process arguments.