No worries about the late response. Are you going to build a box to put the carb in to make up for the difference in pressure? If you are using a carb system just to take it apart later a pull through might be easier. Does building your own ECU save you the majority of your cost for a fuel injected system? I don't have the coding skills for it, so I couldn't pull this off, but from what i have seen of small engine EFI, it seems very pricey. As far as an aftermarket carb, for something temporary I have found the PWK style carbs on amazon to be pretty good. Something like this should work pretty well. You might have to make your own intake, but that wouldn't be all that hard with access to a welder. a 1-1/4 inch FURNCO boot from a big box hardware store to connect them together and you'd be off to the races. Sorry for all the questions, but I find your project o be very exciting.
I am assuming this is a brand new carb and not a second hand carb you bought. If it's used, there is no telling what the last owner did to screw up that carburetor. I say that because I have seen enough crazy stuff from redditors here, who need help, but don't always give us the finer points on the conditions of their parts that had we known we could have solved immediately.
With that said, I suspect you have a vacuum leak. This is a 3 circuit carburetor and the transition circuit takes over from the idle jet when the throttle starts to open a little bit. There is no adjustment for the transition circuit at all and this would be controlling the fuel precisely where you are encountering your problem. That's not me saying that either. That's coming straight from Honda's troubleshooting manual for all of their carburetors and the gx390 carb specifically. That's why you changed your emulsion tube to the smaller one in an effort to change the air-fuel mixture and still had problems and why you are also having trouble at idle and are using so much choke at points where you should be using no choke at all. Your current carb setup is a lot worse than your stock carb. Rich mixtures just cause fouling, which while not great is preferred over lean. Lean mixtures that get way too lean can cause detonation and that will ruin new engines in a hurry.
If I am right then we absolutely have to fix that first before you fiddle around with the carb anymore. There are 3 places you could have a leak, and I am not sure which one would be causing this specific issue. There should be a plastic piece with an o-ring between the carb and the airbox. That has to be on there or it will screw up everything on the carb flow for sure. There is another gasket between the carb and the black insulator block. That one is also directional and there is a little hole that must be on the same side as the outside carb stud and not the stud closer to the flywheel. There is a final gasket between the insulator and engine. Just make sure that it is present and in good shape as well. I suspect that your problem is with one of the first 2 gaskets, but you never know. If holes are not blocked by the air cleaner gasket that need to be or holes that shouldn't be blocked by the insulator gasket are, then that would certainly explain the tuning difficulty.
The stock emulsion tube should be working with that carb. You shouldn't have Frankenstein mix and match to get things working, and that's not a recommended fix to begin with assuming that's a brand new emulsion tube. The only other thing I can think of off hand that could be wrong provided that you don't have a vacuum leak is that you have the wrong pilot and main jets for your elevation. However, having installed a new carb on an engine that I couldn't get working initially, I can tell you that I screwed up my gasket between my keihin slide carb and engine and it made diagnosing the issue a bitch-and-a-half. My carb came with jets that were too small, but mine was meant for 2-stroke dirt bikes, not the gx390. That was easy to figure out once I went through all my jets. I'd like to save you that headache if I can. What you're describing definitely sounds like a "not enough fuel /too much air problem" for sure, so I'd make sure no air is coming from places it shouldn't be. Especially since you have made some wonky adjustments for air to solve your main jet mixture problem, and you are out of fuel adjustment for the idle circuit.
On a side note, if the mikuni is too expensive, I did have good luck with a Goofit brand clone of the Keihin carburetor. I got mine on Amazon for around $30, and it got my 420 cc engine running like a song. just make sure you buy a jet kit to go with it, which is less than $10. I don't remember the exact price for the jets offhand. I think a 28mm carb would be the best bet for that size engine. This pwk clone looks promising. It would provide you with a lot more fuel without sacrificing your ability to fire up cold or stay at idle. There is such a thing as too big. The GX390 carb is the largely recommended upgrade for that engine, because it's a 3-circuit carb. I am just suggesting this in case you'd rather have the response of a slide carb instead of a butterfly carb. You will need to get new intake to mount that carb, but I don't know the exact dimesions for the gx270. If you go that route you are on your own for that part. Sorry! Good luck, and let me know if any of this advice helps you solve your problem.