I already have any amazing opportunity for one that opens up in September. Ideally I'd have a visa figured out by then

Quite certainly impossible. Don't know if you can rent a flat with a tourist visa, probably not. And even if you can, you can't apply for a work visa inside the country as a non-EU/EFTA tourist.

and have them provide the information about not being able to find another viable candidate in Switzerland or the EU?

Sounds easy, but it won't be. Depending on the occupational sector, it can take up to years to prove that you have searched unsuccessfully for potential candidates within Switzerland and EU/EFTA. Besides, without a consultant/lawyer, who will probably cost a lot of money, you usually won't stand a chance anyway.

I have another connection that works with helping refugees immigrate to the country, as an American is there a way to be a non-disposable employee in that vein?

I don't know if this could be possible, but I strongly suspect that it is not. Unless perhaps it is an international non-profit organisation. You would have to contact the company in question.

I'm also a musician, which I know there's no shortage of especially in Basel but could a case be made for uniqueness required there?

That, too, will probably be impossible. However, it is best to clarify this with the cantonal authorities.

As far as I can tell, the only possibility in your situation of living in Switzerland is to apply properly for a job according to https://www.sem.admin.ch/sem/en/home/themen/arbeit/nicht-eu_efta-angehoerige.html .