I think this is where two things might be useful for you 1. take a really deep breath and 2. try and consider this from your son's point of view.
In an ideal world - wouldn't it be great if we lived in one of those - your son's mother would have discussed this with you and between you both a suitable arrangement might have been reached. But and its a big but, that's not the space you are in unfortunately.
So consider this from your son's point of view. He's having a ball (hopefully) at camp and is looking forward to spending time with his friend but camp finishes and Dad is there to collect him telling him he's going to have a great time with Dad for the week. How do you think he'll react and how will that reaction whether expressed or internal ultimately effect his relationship with you?
Like Clubman I've been there, and I can tell you 13 is a very tough age. No matter how you try and shield them from the ugliness of a breakup they see so much more than you think. So I'm going to suggest you consider an alternative:
Leave things as already arranged, when you do see your son be positive about him getting to spend time with his friend, engage with him about what he has done and be positive. Trust me when I say that is what he'll remember. I'd put money on it he's dreading upsetting you, getting mam in trouble, getting you in trouble, upsetting Mam and god knows what else. So make it easy on him.
Post holiday you need to consider communication with his mam going forward and I speak from a very similar position to you. I would highly recommend you consider a co-parenting course. Both I and my kids dad were advised to do one, he refused but I did it anyway and I'm really glad I did. It doesn't solve situations like your current one but its does help you refocus your mind to what you can do and ultimately vastly improves your relationship with your children. It also helps you deal with the other parent in a manner that takes all the heat and fight out of the situation. Its very hard for anyone to continue a fight when one of the two people refuses to engage in fighting.
And that deep breath, its helps, don't make knee jerk decisions, take time and a step back from the situation and pick what makes things easier for your child. It may seem "unfair" or "one-sided" but trust me your kids will respect how you have behaved - they do eventually turn into adults and will understand so much better what and why you have acted as you have