Well anything can fly at 5000’, even a commercial jet, they just use a lot more fuel and go a bit slower.
Fixed wing is definitely better for range and speed at any altitude, the question is just which one to choose and how much do you want to pay?
It’s very difficult to get accurate data for this type of thing as the maximum cruise speed quoted in places like Wikipedia may be either optimistic, based on high altitude performance, or both. Also the quoted max range might not be attainable at the quoted cruise speed and/or the altitude you want to go at.
I gave figures for the Dash 8 and Shrike Commander because I happen to know them. There may be other aircraft that are more suitable but it helps to have the flight manual available to check the actual performance you can expect for what you want to do. Helicopters on the other hand are designed for low level ops so the performance figures quoted in places like Wikipedia are more likely to be relevant to your scenario.
Giving this some more thought, the best way to get the most performance out of an aircraft doing what you want is to fly in a pressurised aircraft at the highest altitude it can operate at with the cabin pressurised to 5000’. Using a Dash 8 again as an example, just because I’m familiar, not because it would be the best option, you should be able to fly it at 22,000 feet which is only 3,000 below its service ceiling. You could then do it in 8.2 hours but it is just beyond the max range. You could get there but you’d be landing on fumes which is illegal of course. That would be using a Q200 model. There are a small number of Q300 models with long range tanks fitted to the cabin. Using one of those you could go all the way in one hop but the Q300s fly slower so the time taken ends up being the same.
Q200 8.2 hours flight time plus time for one stop, call it 8.9 hours.
Q300 8.9 hours non-stop.
Something else to consider, the Q200 aircraft are readily available while the modified Q300s are not, there are five of them and they’re all in Australia.
Finally, if the aircraft had a pressurisation problem you may end up, depending on your medical problem, being a dead man. Given that the Q200 can do the trip at low level, unpressurised, with one stop in about 10 hours, that’s what I’d be doing. That’s an extra hour transit time with a guarantee that you arrive alive.
This is all assuming you can actually fly LA to NYC at 5000’, I have no idea what the terrain is like between those places. If you can’t do it at 5000’ in a straight line then you might have to go with the pressurised aircraft flying with the cabin at 5000’ and risk possible death.
On preview I see that Omphaloskeptic has addressed this.
It would be interesting to find out the low level performance of some other commercial aircraft.