FWIW, I'd recommend paying close attention to the nuances of maizeman's 9:22 post. This is not just a question of whether you inform your joint supervisor -- if this guy perceived you as going over his head to his boss, that will damage the relationship more than you just laying down the law yourself and arguing it out. He has committed himself to a position, and so you proving you are right and he is wrong will provoke hard feelings, period; forcing him to be wrong in front of his boss is even worse.
You need to find a way that allows him to save face, if at all possible. I.e., give him an out, give him a way that he can say, "oh, sure, you never told me about *that* -- that changes the issue entirely." I like the idea of the email where you are very complimentary of his experience, and then point out the things that you have been told *by your supervisor in the clinic* you need to do, and that this puts you in an awkward position, because you see his point but feel stuck, and so you'd appreciate his help figuring this out -- would it make sense to set up a joint meeting with [joint supervisor]?
Do not, under any circumstances, run around and curry support from everyone else for your position behind his back so you can then demonstrate in front of everyone that he is wrong and you are right. That is a last resort if he will not comply, because it WILL generate hard feelings and WILL be held against you. Try to involve him in the fix, be open and respectful with him, and avoid embarrassing him if you have any other way to fix the problem.
The thing to realize here is that a successful career requires both substantive knowledge and people skills, and the more difficult the people you are working with, the more finely-honed your people skills need to be. So even though you have the superior technical knowledge here, look at it as an opportunity to build your "soft" skills -- it will serve you well in the future.