[embee]

Balancing in theory can be done with or without the airbox, at idle the effect of the airbox on intake pressures will be negligible.

Idle mixture adjustment is best done with the airbox on if at all possible. On the curvey SV650 it is very difficult to get to the mixture screws without using a special right angle drive tool, though it can be done.

I have found a thin thermal glove will let me have my hand in there just long enough to tweak the carbs with the engine hot. Front one I use a long thin electrical type screwdriver, rear one I use a small screwdriver bit in a 1/4" drive socket and hex adapter, you have to develop the tools for yourself, what will fit and what you can get your hand in with.

The rear one is a lot easier to get to from abovee with the airbox off, and I've found it makes very slight difference to the final adjustment, so is probably the easiest way. You may find the screws siezed, in which case remove the carbs to sort it out on the bench.

Carbs often benefit from having the floatbowls removed and cleaned, water/rust powder inevitably collects in there.

Also check the choke plungers for closing off properly, corrosion often is seen where the cable meets the plunger and it van easily stop them fully seating, which causes all sorts of poor running. I has this on my bike after only a few months from new.

Sounds a bit to me klike you could have some dodgy ignition item, maybe a coil, maybe just connections, maybe plug cap or HT lead.

I'd recommend iridium plugs as they will tolerate more mixture variation than "conventional" plugs, though it sounds a bit more fundamental in your case.