You are severly underestimating how long it will take you. Even if you're averaging 70 mph, when you factor in the stops, fueling, meals... you will drop to around averaging 50 miles in an hour. It takes 20 hours with horses to get from Chicago to Denver. On long hauls by myself, I can do 800 miles in 16 hours but that last hour is a blurr and I wouldn't recommend it. You never get out on the road when you need to either. Plan on a 2-3 hour late start. I'd break it down into 1 1/2 to 2 days mostly because both you and your horse haven't done it before. County fairgrounds are a great place to stop for the night. You can even just stop there for a few hours and grab a nap in the parking lot. Open the windows to the trailer for some good air flow but leave him in it.
Have some banamine on hand just in case. There are sections of I 80 in W Nebraska and I 76 in E Colorado that are very lonely and you can have 40+ miles between fueling. Carry a full fuel can with you. Your mileage will also be a little lower than normal because you are doing some serious climbing eventhough you can't tell it (plus there's a major headwind). In W Nebraska and Kansas, there are "farmer" gas stations along side the interstate in the middle of nowhere. They're gynormous so they can just drive the combines up and fuel up but they are also sweet for horse trailers because you know you will never not fit into one of them and they're easy off, easy on. Wish I had all the exits for them written down. Heading W out of Grand Island, there are some nice rest areas. Tree lined and a little bit off the interstate, they aren't a bad spot to eat a sandwich while he chills out in the trailer. (and clean!)