Starting a startup is starting a company. Starting a company is a business thing, not an engineering thing. Think of it as being able to sell a product, before you even have a product to sell. You cannot really operate 'as a company' without having an actual licensed company. You may need to do a bit more leg work before you start this venture in earnest.
Then reach out to the enterprise. You have to know exactly who to route your request to in the company; you can't just talk to "sales" or "customer support" as those people have no idea how to talk to their own engineering groups. You'll need to know the language that the particular group in an enterprise speaks, so they will actually consider your message, prioritize it, and do some work on it. Cold-calling generally doesn't work unless you are very good at sales. You will probably need to engage with someone at the enterprise on a different pretext, and slowly shift the focus over to the thing you actually want, and hope whoever you're talking to will do some leg work for you to get you to the right group.
Try to position your request as an offer to do something for them. I am building a product that delivers your product more customers, I am building a product that supports all of your competitors and figured you might want to be supported too, I want my development team to work with your API and give you feedback about how well it works for us and report bugs to you. But don't make it sound like a sales call. Make it sound like somebody in the company already expects this and you're just trying to coordinate with them on something they already want to do.
But if they actually don't want a lot of use cases right now, you may have to position it as a test case that isn't being released to new customers, so they don't have to worry about building out their production support infrastructure and being on the hook for it going down. SLAs and EULAs are also a concern. Then of course there's data privacy, regulations, etc to consider. In many cases you may need to establish a legal contract in order for both parties to be covered.