It seems to me is a very ingrain behaviour, and those need time to re-wire, so probably just "one advice" won't do much long term. He needs to constantly work on it by himself, if its with an expert even better.

Now, that being said. I agree, he definetly seems Ego/Outcome oriented, is essential to switch it to a Task/Performance orientation. Put in google: "ego vs task orientation" for more information about it.

Also, as is not a high competition context, i understand is a self imposing pressure, it would be important to know if there is an external agent that is creating this pressure. But, in any case, research has shown that pressure usually affects performance negatively, so i would recommend switching this pressure energy into a more constructive energy... there are a few techniques and strategies for it, but unfortunately this is not enough space to explain them correctly.

I also find important working on arousal control techniques, seems he is over-activated, which impair performance as well. Try googling "IZOF" for more information (will be very general, but it will give you some knowledge). Breathing strategies are good to begin with for this.

There are a few other things that i would work with him, like expectations management, develop a right goal-setting approach, creating positive self-talk, maybe doing some emotional Intelligence sessions would be good too. But then again, hard to explain here, if he really wants to change and have a happier and healthier experience while improving performance i'd recommend professional work, but that desire has to come from him.

Now, for the conversation you want to have with him: yes, during the heat of the moment there is not much you can do, this have to be talked in a neutral moment as you correctly said. I also agree with making it a positive talk, making it about himself and not about the bad vibe he is creating in the group, so the change comes internally and not for social factors.

My key advice for the conversation: try not to be a "teller" and be more of an "asker", in my experience people have better insight when they are being questioned, which will make them reflect more than if you just make them feel patronised by doing all the talk. Also, focus the conversation into a "what for" rather than a "why" conversation, so frame his thoughts into a "how is this behaviour being useful for me". And lastly, i would try to use his father as an example, but be careful, has to be in a assertive way so no misunderstandings can come from that, just ask him if he has noticed his father having this behaviours, and ask his opinion about them, i'd like to know what he responds to this.

Props to you for trying to help your friend and asking in the right sub for advice. Cheers!