TL,DR: Be sub to him, but dom to the kingHow do you get Charles to be your mount? I remember not being able to figure it out. There was no message in his dialogue menu and I loaded a save to before Bradley was my mount but I still couldn't get it to work. Hopefully someone's figured it out a month later.
When you first met him, give him a bj, at the next encounters, always accept to do a bj or anal with him.
After making the king submissive, ask him to give Charles as a ride.
You need to be versatile to make this work.