I believe pyrethrin is the natural one, while permethrin is a synthetic. YOu can google search. Can you use a shovel to flip the soil where the infestation is to help them reach the bugs. What are root weevils? Got a pic? Where are you???
OK, I did a google search. What plants are they infesting? If the infested plants are ones that you are willing to let the chickens continue to work on, and if the chickens have daily access to the plants, you might wait it out and see what happens. They could give you all the "fix" you'll need. But, if the weevils are killing a prized ornamental or food crop, you might want to treat with one of the p...thrins. I had horrid Japanese lily beetle infestations. To the point where I'd given up on growing them. Chickens moved into that bed, and there was not a single JLB there this year. I bet those little larvae are pretty tasty, if you're a chicken!!
ETA: correction re: permethrin/pyrethrin.