Your question is pretty abstract, so my answer will be abstract too: if you're ready to own the negative consequences of your action should they arise, then take the action, otherwise don't. Whatever arguments you read here, they won't be a valid excuse for you later, it's on you.
This seems abstracted enough to make it impossible to answer, as it depends on the nature of your relationship, the action, and the harm. E.g. if you are risking direct bodily injury to P, then he would have much more of a right to veto it than if your action was something like "I will carry out research into a controversial topic which might end up upsetting other people and make them angry at me, and also angry at P by association since we are known to be friends".
But in general, would you want someone else to unilaterally carry out an action that you felt had a high risk of harming you, just because you failed to convince them of this being the case?
I think you are focusing on the wrong aspect of your proposed action. The question is not whether you owe it to P to accept their class arguments, the question is whether you owe it to P to be open about doing the deed anyway. While I think you do not have a more obligation to listen to P's argument, having had this conversation with P, you are morally obligated to tell them ahead of time that you are going to do it anyway. Going behind their back and hoping they never find out seems like a betrayal of their trust / lie of omission. Having told in strong terms that they are against the action, P can reasonably expect you not do it behind their back.
1. Any good answer depends on the action.
material risk
I'm not sure what you mean by this phrase.
2. Absent the object level...If you ignore 1, then you should ask, what will you do if one of these 'Bad Outcomes' occur?
I wish to perform some action which invites material risk upon myself and another person P, as well as inviting benefits. My personal evaluation is that the benefits far outweigh the risk, but the risk is both speculative and partially subjective so I decided to consult P before taking the action.
P was vigorously opposed to the action and gave his reasoning, a list of factors that he claims suggests the risk is much higher and the benefits much lower. Some of the factors given were of the inherently subjective variety, such as feeling proud of the current status quo.
Afterwards, I sat and thought about these factors and still reached my initial conclusion that the benefits far outweigh the risk. Furthermore, not only can I perform this action unilaterally but P will not even be aware I have performed this action unless one of the Bad Outcomes (which in my evaluation are exceeding unlikely) occur.
If the risk were greater than the benefit I would not take this action but P failed to convince me this is the case.
My question:
If my analysis is correct, with overwhelming probability not only will P be unharmed by my actions but also be entirely unaware of them. With this in mind, do I have a moral obligation to return to P and argue my case and obtain consent before performing my action?