3 years ago, I tore down a ice house. The walls were filled a foot thick with saw duct. I was totaly surprised by the lack of mice nests. I then smelled the sawdust, even after 100 years, you could still smell the cedar.
As long a the sawdust is kiln-dryed and not of green wood, you should be fine. They are building houses with straw walls, and they don't treat them for a fire hazard. The best, I think like the old ice house, if you can find cedar sawdust, that should eliminate any insect or rodent problem you might have. I have been using sawdust around the farm since the mid -1980's, mostly for bedding, But I have insulated water tanks, well heads and misc. buildings with it for years. One prblem is spontaneous combustion, but the saw dust has to be soaking wet or made from green wood. Saw dust like most loose insulation will settle over time, you may want to be able to reload the walls later.
One particular thing about sawdust, especially kiln-dried sawdust. It absorbes moisture. The neighbors' cattle barn, used to have the walls dripping wet in the winter from the expiration from the cows, when he used straw. When he used sawdust, all of the wetness disappearred. His cows suddenly became much healthier.