How can I ask/negotiate to work permanently out of state in a way that both conveys urgency and preserves my option to stay if they say no?

Even if this situation wasn't happening, you don't have the option to singlehandedly choose to stay. Employment is a two-way street, and your employer can dismiss you (while honoring the relevant notice) when they want to - just like how you are able to quit when you want to

While I do not have a concrete job offer, if the answer is "no" I will strongly consider leaving the company and I have recently had several realistic opportunities to do so.

This is pretty much the default assumption made by most (if not all) companies when faced with an employee who out of the blue ask for a significant change in their employment that mostly benefits the employee.

It is reasonable for an employer to assume this. If you're asking for something more, that means that you either (a) need it or (b) feel that it is fair compensation for you. In either case, not getting what you ask for means that you're likely going to look for other opportunities where you get it. You might even already have opportunities lined up and could simply be checking if your employer is willing to match it.

I have proven to the company that I can successfully drive big international projects while working remotely. In many ways, I have been more productive working remotely.

As much as I sympathize with this position and like to think I have established a similar reputation, that's not the right way to compare today to the "back to the office" period.

Even if you get your way, everyone else will be going back to the office. And instead of being a more productive remote worker compared to your colleagues, you're going to be compared to what your colleagues can do when working together in the office.

Your remoteness is going to add an overhead cost. Hybrid meetings (i.e. some in person, some remote) tend to flow more awkwardly than fully online or fully in person meetings. Intentional or not, the re-emergence of hallway conversations is going to lower the amount of communication via public or official channels, which means that you might start being out of the loop on smaller issues, which in turn can lead to repeatedly being caught unaware of things everyone else knows, which can affect your reputation.

Your question consists of justifications of how remote working has positively impacted your work, but that doesn't necessarily mean that it wouldn't negatively impact others. Your company is likely to be interested in the total sum more than your specific performance, and will take those side effects into account.

I'm not omniscient and I don't know if you will face these issues. But neither will your company, and they will still be forced to make a judgment call before being able to accept/reject your request of going full remote.

Even if they agree, it's possible that the company will now actively seek to replace you with someone who is available for working in the office. You cannot argue this away. No matter how much you prove that you're more efficient working remotely, the company is free to look for an in-office employee if they prefer having in-office employees. Both you and your employer are free at any time to look for a replacement employer/employee.

Assuming that I am accurately conveying the situation, what is the best strategy for how to bring this up to my boss? The reality of my company is that he will not have the authority to say yes, but will have to go to his boss's boss, and possibly a level or two above that.

This may work to your benefit, if you can frame it correctly. If you ask your boss on a personal/friendly/casual level if he's aware of any plans the company has for remote working when the pandemic passes, he may be able to relay information without needing to broadcast that you are seeking fulltime remote work.

Even if he doesn't know, he might ask around without particularly putting your name in the spotlight. But it does require you to have a good personal relationship with your boss.

I do not want to give an ultimatum, because if they say "no" I want the option to stay a couple of additional years without poisoning the relationship. But I do want them to be at least mindful of the reality that if they say "no", there is a decent chance that I could leave very soon.

Imagine telling your boy/girlfriend that they're not allowed to date anyone else, but you're going to still be dating other people and might dump them if a better boy/girlfriend comes along. That is effectively what you're doing here.

That is simply not how it works. When one part of the relationship loses interest, generally speaking so does the other. Your company is not pining for you.

I am more comfortable discussing the case for the company as to why this move will be good for them

A vacuum salesman will tell the customer how good their life would be if they purchased this vacuum. And the customer is naturally deincentivized to believe this, because they know the vacuum salesman is just trying to push sales. That doesn't mean he's invariably lying, but it heavily taints the authenticity of his message.

It might be better for you to not spend time arguing the company's case, so as not to make it feel like you're manipulating the situation. Especially if you don't want to taint your reputation.

If you leave the choice up to them, it may be better as your request comes across as an open question rather than a done deal that you're asking them to sign.

But where I struggle is in conveying the urgency, on their part, of making this decision. How can I do this?

Urgency entails a high necessity. As established before, a high necessity entails a likelihood of you quitting your job when you don't get the thing you "urgently need". Phrase it however you want, any company that can reason about its own decisions will understand the nature of your request and the likelihood of you not staying with them when that request is denied.

At the end of the day, no one can say how your company will react. Maybe they'll be happy to accept. Maybe they'll abjectly refuse any long-term remote work in a post-pandemic world. Maybe they're willing to seek a compromise with you. Maybe not.

But what I can say is that you're going to have to change your expectations on specific you can be about this request while at the same time staying completely free from any consequences if it fails. You cannot have this cake and eat it.

The company might possibly let you have the cake and eat it (if they genuinely don't want to replace you), but you cannot guarantee this in advance.