UPDATE: Success!
Since my last post we have not had one rat. We've gone back to our original method of feeding but can reevaluate that if we see the problem again.
We had a major infestation. They were scurrying around the run all day (babies and big ones). We saw them weaving around the chickens feet. We learned that our chickens are pacifists! They were happy to share their food!
The reason I believe we were successful with the poison is that we laid a LOT of hardware cloth aprons all around the inside and outside of the run (our coop is housed in the run). The rats that were outside couldn't get in very easily and the ones that were inside were basically stuck.
We fed them only wet food (made less mess) twice daily removing it after each feeding. Therefore the only food available to the rats that were stuck in the run was the poison. The first night it went pretty much untouched but by the second night they were getting hungry and gobbled it up. We literally haven't seen one since. The hardware cloth took a few days, but after that it took two days to eliminate the entire problem.
We never saw any dead rats, they must have all died down in their "hidey holes".
The poison we used is Tom Cat, we got it at HD. I did buy one of the bait boxes that is outside my run (although it hasn't been touched) and in the run (where the chickens cannot access) I screwed down a few of the bricks to a scrap piece of wood (the bricks already have a hole through the middle). The advantage of doing this is is:
1. The rats can't drag it down to their tunnels preventing others from getting it.
2. You can see easily if the bate is being taken.
Happy to go into more detail if anyone else is struggling with this problem . I'm no expert, but given our situation the above worked. We were very anxious to get it taken care of especially because our run/coop is against our house and eventually the rats would have made their way into our home.
We continue to monitor the problem and will make changes as needed.
Thanks to all for your help along the way!