I am not familiar with Idaho's game laws but I don't live there. Since you live and presumably hunt there, you should make sure you're familiar with them. I did a quick internet search for this topic and came up with the link below. That's a start. It looks like dogs are prohibited from "actively tracking, pursuing, harassing, attacking or killing deer or any other big game animal" in the state. Based on your statements above, it sounds like you should definitely call Idaho Fish and Game and ask to speak with a warden. They should be able to answer your questions more in depth and help you avoid problems.
https://idfg.idaho.gov/question/dogs-chasing-deer