I’ll make one important assumption: That you didn’t fiddle with the balance before you started tweaking a more or less properly adjusted system.
If that assumption is true, then any procedure that starts by having you tweak a setting that “wasn’t broke” is well-intentioned bad advice (even though it might be a fine procedure to use from any other starting point.) You needed to fix your tweaks without breaking something new.
I’m with Bill. The balance settings are in most cases pretty much for ultra-fine tuning, or when someone has tweaked a throttle shaft. Rohan is on the money as to their importance in this case, but I suspect it doesn’t fit your use case. However, if you have a musical ear and a length of rubber tubing, you can effectively set balance by equalizing the tone of the hiss when the tube is placed in front of each throat.
Shade-tree adjustment of Webers or Dells (once you’ve elected not to mess with balance) is per the book: Make minor incremental adjustments to each mixture screw until you get maximum RPM – at this point, any change to any throat lowers RPM. Set the idle correctly, then sit back and bask in your accomplishment, because at this point, you’re done with the fixes a screwdriver will allow.
I’m intrigued by the notion of setting idle speed up. While it does provide a bigger scale to play with, it also runs you closer to transitions where the progression holes come into play. I hadn’t heard this suggested before. It is important to remember that idle mixture screws only affect the behavior of the engine while running on the idle jet (typically below 3000 RPM), not the rest of the circuits. For example, you can’t make your engine run richer above 3000 RPM by playing with screws. Trying to do so will just mess up your idle mixture to match your main jet or transition mixture problems.
Personally, I find that starting with low idle RPM it is much easier to sense the effect of turning the mixture screws than at higher idle. Preference, I suppose.
To your original problem (and the one you’re back to), I suspect it had nothing to do with mixture screw settings. On these carbs, the screws represent a very small range of adjustment in a sea of overall tweakability.
I would carefully clean and check the jets for the “weak” throat. If there is lean or rich condition after getting all cylinders to pull their load, I would then swap the appropriate jets and air correctors. This game is expensive unless you have a friend with a drawerful because you need four of everything, but I believe it’s what most carbs that need significant tuning really need. Disregard this if you know that the system was tuned by a recognized expert with a dyno and the inlet and exhaust plumbing has not changed since that time, in that case you might want to consider new jets of the same spec for any problem throats.
Advice for the next time around: If you know you will be adjusting mixture and balance screws, start by identifying to the nearest 1/8 turn where each of them is set (these pictures are easy to draw.) This information will suggest (if there’s a large difference between the settings for any cylinder) that the previous tuner had some struggles with an abnormality in your setup or not. Then you have to decide if he was dealing with a real discrepancy or was just a poor tuner.