I wanted my step son to bring my son on holiday with him. I even said I'd pay.
Could you share with me why you want this?
I want them to get along with each other.
That's very valid and I hope they get along. However, your step-son taking a vacation without your son doesn't make that impossible. Consider that they just wanted a vacation on their own.
It's not my fault that my step son doesn't like to travel with the family. He is my step son! We all have to put up with each other!
Perhaps you could plan a day where you, your son, and step-son spend it together. Make it a day where you are all available and it's something each can enjoy.