There is that "intent" word again. From an insurance standpoint, it would seem that they intended to block the road, but didn't intend to hurt someone. It does seem as though they did not act prudently, and they were, therefore, negligent. If they have a liability policy, it would probably have to pay out assuming bodily injury to the child (which does not seem to be the case).
Yes, it would be a negligence action, and the carrier would certainly have a duty to defend and a duty to pay if the court rendered a final judgment against BLM that was upheld. However, I'm not sure they'd go that far. Let's put it this way. If they went to trial and lost, the carrier would probably appeal and ask the appellate courts to effectively derecognize the cause of action (a legal insufficiency the evidence ruling, a ruling that BLM had no duty to order is protestors not to stop traffic, etc.) before paying out.
However, I seriously doubt any company would insure BLM.
Lol. Can you blame them?