Hi,
Sorry you're having troubles. These.are known as really great outboards so I know how frustrating this would be.
I don't work on these either so no "silver bullet" for a fix although I do have a good understanding about the "vagaries" of these little Keihin carburetors.
Just some ideas about things you might consider and some moral support.
Hopefully you're working with a shop manual as I don't have a copy to consult. I do have the Honda Marine Carburetion Manual.
Hope you don't mind some questions but me asking sometimes gets an owner to come up with the solution.
Did this problem start suddenly or develop slowly over time?
Did you synchronize the carbs after cleaning them?
Have you ever checked/adjusted the valve clearances?
While the carb sync and valve clearances are important, my first impression about what you're describing is that there could be something going on with cam or ignition timing.
The beauty of checking/adjusting valve clearances is that the cam timing components, (belt, tensioner) are accessed while doing so. This engine has a spring (item 8 in the link below) that maintains belt tension and, if that spring gets weak, the timing belt can go slack. I would want to look at that for a complaint like this.
As far as ignition timing (as opposed to cam timing) that is a function of the pulser coil (item 10 in the link above) sending a signal to the CDI (item 1 in the link below) and the internal "mapping" built into that unit to adjust the spark.
I can't tell you how to test either the CDI or pulser coil but the service manual will have something about those. Although how useful that info might be, I'm not sure. BUT...there should at least be a "pass/fail" resistance value for the pulser.
Since the engine starts, idles and powers up to 3,000 rpm, I tend to think that the pulser and CDI are probably ok.
That brings me to consider electrical connectors.
In cases like this I always suspect engine vibration and the effect it can have on electrical connectors.
The harmonics that develop with vibration can cause a connection that isn't secure to sometimes act almost like a switch. The fact your engine runs up to the same rpm then starts crapping out followed by runs good when you drop below that "trigger point" might be an indication that something like that is happening.
Because of that, I might be inclined to check all electrical connectors at the pulser coil, the CDI, the ignition coils and the exciter (power) coil (item 1 in the link below)
I would also check the integrity of the kill and ignition switch connectors as they can also be affected by the same vibrations.
There are other things it could be....weak valve springs... or... even something missed in the carb cleaning. But I wanted to
give you some ideas you could check without a great deal of disassembly and or testing.
I hope you can figure this out before giving up on your Honda and I'm always glad to discuss the problem if you think that would help.
Good luck.