Getting the test...not a problem...getting the results within three days of the flight...don't think that's going to happen without some amazing US-sized "rush testing" charge. And, in the US, the sky's the limit when it comes to how much you would be charged for such a thing, as in several thousand US dollars- hell, that was what some people have been charged for a regular, non-rush test. And in NY...would be extra-special expensive.
I really feel for you- I am about to immigrate to Taiwan and, at this rate, will have to do the same thing to satisfy Taiwan, only I am lucky enough to be in HK where it's a lot faster and cheaper.
I like your idea of going to another country and getting the testing...except that many countries are less than thrilled having Americans show up. Not sure if Canada is still letting in Americans but they have a ton of flights to HK (well, maybe not a TON these days, but you could get a flight if you head to a major city like Toronto or Vancouver). Pretty sure Cathay or Air Canada is going to let you on with your negative result from Toronto. But, of course, that's a few days of hotel in Toronto...balance that cost against the cost of rush testing in the US... if you can pay to get it. Have seen reports where test results are coming back a WEEK after the test, what a mess. Best of luck to you, keep trying, hope you find something.