I was over that road in August. It is one of the most spectacular drives you will take. You will not enjoy it towing. If your 40' rv is a gas engine, you absolutely shouldn't try this. If diesel with good engine brake, you could do it...but I wouldn't.
PS...I find many people who are far too cautious about "climbing mountain passes etc." I'm not one of them, but this is a pretty serious road.
Stay in Red Lodge and drive the toad...there are two options, and the KOA wouldn't be my first choice. Red Lodge is a great little town. Or stay in Cody and drive the toad. Cody is also a great little town.