I have just balanced my bike's carburetors (they were initially way out of sync). It idles lower and seems to smoke more when I rev it (smells rich). Returning it to it's previous out of sync setting increases the idle rpm.
It's an 2004 Suzuki GSX250 Invader (2 cylinder)
I'm not an expert on bike carbs, but if they are the same as car ones there will be two settings - one for mixture and one for idle. Once you have them balanced, you'll need to adjust these to get the right levels, while still keeping the pair balanced (i.e. increase the idle on both by the same amount, then check the balance, then decrease the mixture a bit, check the balance again, and so on).
In my experience it's something of a dark art, and you'll probably spend ages adjusting them by tiny amounts them until you're satisfied...
Multi-cylinder vehicles with multiple carbs
To port this question over to applicability with cars. This is the same issue that has plagued older Ferrari and Jaguar mechanics over the years. The 'balancing act'. If you have a V12 and 6 or 12 carbs, getting them all balanced and uniformly restrictive in terms of allowing air into them can be very challenging. MOST multi-cylinder motorcycles have 1 carburetor or throttle body per cylinder.
Suzuki GSX250
From the factory, your fuel screw is blocked by a small aluminum plug. You cannot adjust air/fuel ratio (AFR) on this bike until you drill out the plugs. They are very easy to remove. They are dangerous to drill out. If you penetrate through the aluminum plug you can push your drill down to the brass air adjustment screw and turn it all the way in. There is a needle at the end that will enter the air galley and get stuck. When you try and turn it out the pin snaps off into the air galley which provides proper AFR at idle and just off idle, since it's an air galley, it will be rich on that cylinder.
AFR and carburetor synchronization
They do interact but have very little effect on one another. That being said, if you give one cylinder a super lean condition and the other a super rich one you can effect the synchronization. As well, the inverse is not true. If you have one carburetor set open too much and the other one closed off, their AFR won't change.
Concern
The real concern here is that at idle and when synchronized the bike doesn't run good. When you balance by ear it seems to run smoother. This is indicative of another issue. You mentioned white smoke, that's oil. Your theory that the air filter is too restrictive could have merit but honestly, it's an outlier statistically. Check the air box before the filter. If you have a bunch of oil in there then it could be that you have a bad PVC valve in your valve cover OR excessive blow-by due to bad rings.
Conclusion
Based upon the information provided here is what I believe. You have one of your cylinders that has bad compression. That's why it runs unevenly when the carbs are balanced. (Quit thinking of AFR right now) If one cylinder runs more efficiently than the other then it will be out of balance. It being the engine. When you are synchronizing the carbs by ear and making the engine run smoother you are compensating for the poor compression of one of the cylinders.
You can do a compression test with a compression tool or you can perform a leak down test to see where air is escaping by listening to the PVC for sounds (bad rings), the exhaust for sound (bad exhaust valve adjustment, burnt valve, bad valve somehow, or listening at the intake (bad valve adjustment, bad valve).
The oil source for the smoke could be bad valve guides, bad valve guide seals or bad rings.
