Sounds like an annoying issue, and I agree with your thoughts that it sure sounds like you're sucking the bowls dry. First thing I would do is try to confirm that's what is going on.
Assuming you don't have the early carbs with the bowl drain holes,... I would let the engine idle for a minute or so to make sure everything was stable. Then shut off the engine and drain each bowl into some sort of small graduated container, and measure how much fuel was in each one.
Then I would make your WOT run until your O2 sensor went lean and then quickly kill the motor (coast to a stop) and then do the same fuel volume measurement again. In theory, if your fuel supply system was able to keep up under high load conditions, you should have about the same amount of fuel as when you tested it at idle.
If you get very little out of the bowls, then you have at confirmed the root problem of fuel starvation.
You could do the same kind of test with the clear tube "Teed" into the bowl outlet, but I'm not sure I'd want to go driving around like that .
Other thoughts?
I once had similar problems, and traced it to debris partially clogging the needle valve. I was running from a small engine compartment fuel source and I was frequently wiping it clean and dry with paper towels. I was not running any banjo filters either, and the fibers from the paper towels were building up at the needle valve until they created restrictions. I know you said you checked your needle valves, but thought I would throw that out there anyway.
And lastly, I don't think it has anything to do with a washer under the suction piston spring.