Pick one of the nights that is a big draw, but not for rode fans, if you can do it. The teeny boppers aren't going to come for the rodeo, it'll be a ghost town during the events, but it will fill quickly.
Or go for one of the weekend shows that start in the afternoon rather than night.
Or better yet, what I do with my little kids, is to early in the morning on the weekends, and they get to see some of the stuff in the different arenas and barns, there's no one around and people are just killing time and it's a kind of cool behind the scenes type atmosphere for the kids. It's not roping events specifically, but it's animals and an up close look at them all that isn't as available when the big crowds and organized events start cranking during the day.