how do you deal with this conflict once it has been decided that this is something that must go ahead, if you feel strongly against the change?
My way of dealing with this is considering what my role in the company is. Am I there as a developer? As an analyst? As a team lead? As a project manager?
If I'm the developer (which I surmise you are), then you never had "decision making power" to begin with. You simply had some decision making privilege that was extended to you because someone (with actual power) chose to follow your decision.
Edit: you mention the three of you are co-founders, but the answer remains the same. You don't have any power, only a majority of you wields any power. In this case, you are not part of the majority.
If you're not holding the wheel, you're not steering the ship and thus cannot override the decision that is made by whoever is holding the wheel. You can ask them (which you did), but they can ignore/deny/disagree with your request. Not your circus, not your monkeys.
You informatively raised a concern, it was ignored. Accept the outcome and follow the plan as agreed upon. You and your two colleagues are there as a team, not as a darwinian competition.
Even if you were right, the other two developers clearly did not think so and will have to make the mistake before they realize it was a mistake. Everyone does what they consider to be the best thing, until they understand that it's not actually the best thing. In some cases, that means having to learn the hard way. This can apply to either you or your colleages, only time will tell.
If you want to be the one steering the ship, you must first be in a position to do so. But if you're unable to deal with things not going your way, I'm apprehensive about your compatibility in a leadership role, as it entails much more compromise than outsiders seem to think it does.
I find that this severely impacts my mental state (either by making me feel less motivated, agitated, angry/frustrated, or what have you), and I'd like a solution.
The short answer, in the words of Disney's Frozen, let it go. That's easier said than done, but it will boil down to this.
Look inwards and ask yourself why you care so much that you're willing to die on this particular hill. Is it because you only want to do things your way? It is because you are unwilling to learn a different way of doing things? Is it because you have a different appraoch to developing and abhor the alternative?
How can you be sure that you are objectively more correct than the others? If you are, then why did you not used that irrefutible evidence to make your case before the vote?
Never lose sight of the big picture: you're an employee of the company, and you're there to do the work the company tells you to do (within the boundaries defined in your contract). You don't get to decide which work you get to do.
Let it go, and accept that when someone else overrides your decision, the consequences of that decision fall on them. You are not personally responsible for the company's wellbeing against the company's own wishes.
I strongly disagreed with the chosen outcome, which was settled based on majority (we're a team of 3 co-founders), and so I'm not sure how to reconcile the feeling of having to do work to support a path I don't agree with.
First of all: you agreed to a vote, you partook in a vote, and now that the decision hasn't gone your way, you're wanting to pipe up about it. At first glance, I would consider whether you're being a sore loser here.
The fact that a vote needed to take place suggests that the decision was not unanimous, which inherently means that someone was always going to "lose". This time, it is you. So you have two options: you accept the outcome, or you don't.
Imagine if everyone who didn't get their way did not accept anything other than what they wanted. How much progress do you think you'd make if the other two developers blocked you every time one of them did not agree?
As a real world example, imagine if everyone in a democracy had veto power over everyone else, and you could only get things done when every citizen unanimously agrees. Nothing would ever get done.
If you do make a fuss, and let's say you even manage to get your way eventually, you will be known as a sore loser and unwilling to work in a team or compromise. All of these observations will be much more detrimental to your career than the minor benefit of being right.