There's really no normal way. I've been on the giving and receiving end of profanity laced arguments (never shouting matches though), passive aggressive rants and everything in between. In this casei simply told him, without too much emotion, that he doesn't make the decisions and I don't appreciate him constantly putting me in a bad position. Then I sent a follow up email to my staff, copying him and his team, that they should do their jobs and not seek his approval, while acknowledging his objections to those decisions. Time will tell I guess how this comes back to me.
My boss has some great stories about the good old days...apparently some arguments in those times ended up with thrown objects, broken glass, etc
I like the very direct approach you took with the email to your team and his team. In our industry, we're always being so passive (not passive-aggressive, necessarily, but that too, quite often) in how we ask for things, how we message things, etc. We're very non-confrontational because, let's face it, people hate being confrontational. It would also kill our NPS, too, which indirectly impacts our bonuses :P.
But, again, I personally would prefer to be much more direct in my communications with clients, and especially my managers when I believe I have a case against their approaches or whatnot, but it just doesn't fly at my level. A lot of clenching my teeth and taking it up the ass .