As others have said, what you are asking for is not reasonable. You are asking them to employ you for the duration of your interview period, which you may or may not perform as well as anyone else they could have hired for that period of time (you may think you are good at your job; there is no guarantee the company agrees with you). If you are not as good as your "replacement value" (the value of someone else who would have been doing the same job as you, all else being equal) then the company has wasted money on you, and that's not what the company is interested in doing. So no, they absolutely won't hire you on a provisional basis, at least not as a paid gig (they may hire you for free, if it's legal and easy to do so, which it probably isn't).
The interview process is to make sure that your "replacement value" is negative, that is to say that replacing you with any of the other candidates available to them would be more costly to the company, in other words that you are the best person for the job. In addition to pure programming prowess, you also have to be able to do things like explain your work, engage in code reviews, mesh with the team on a collegial level (be friendly and stuff), and so on. The LeetCode portion of the interview is meant less to test that you can code, and more that you understand what you're doing and have good problem-solving skills. Any video from a FAANG (FAANG = "Facebook Amazon Apple Netflix Google", basically the top tech giants) company about interviewing will tell you the same thing: What you say during the interview is much more important than what you do, and the company isn't going to get that level of detail about you by reading whatever code you previously wrote (or claim to have written, more on that in a moment).
The problem with open-source contributions is that there's no proof that you actually made them. Even if you use your own name on your GitHub profile or what have you (which you might not do; if you use a pseudonym on GitHub then it's even worse), there's no guarantee that your friend didn't change his name to yours, or even if it is actually your account that your friend didn't write the code and just gave it to you to submit. Not that I'm saying that I personally think you would do such a thing, but that is how the company sees it; there is no guarantee that you didn't cheat, and given the nonzero possibility that you did cheat, versus another candidate who they know didn't cheat (because they took the LeetCode test), they're going to choose the one they have more confidence in, and that's not going to be you. This is more or less why a GitHub "portfolio" is more or less a stupid exercise in software engineering; nobody actually looks or cares, because the information there is more or less valueless except for very rare cases of extremely prolific people.
@Old_Lamplighter, who has mentioned in others of his comments/answers elsewhere that he's autistic, and hence has experience in this issue, gave you a good suggestion: look into local charities and organizations that may help you with your issue. As someone who does not relate with your problem, my suggestion would be to mention to this company that you have some disability (tell them what your disability is) and provide them medical evidence to the effect that you have this disability (doctor's notes, diagnoses, etc), and ask them what accommodations they can make. Don't tell, ask. They will tell you their standard practice for making accommodations for people with disabilities, and you can take that accommodation or you can leave it and withdraw from this interview.