Well it sounds from your symptoms that your carbs are probably fine and it's your petcock thats wonky. Does the fuel flow out when the tank is taken off the bike? Ie not connected to the carbs either the fuel line or the vacuum line? Ie dribbling fuel right off the spigot?
Also check that when you put the tank on the bike to make sure there is no pressure on the fuel selection lever, if I put some pressure on mine, say bending it a little it will pour out fuel. The only thing that is holding back the torrent of fuel is a little spring loaded diaphram. Any extra pressure loosens it up enough to let the fuel flow. Maybe take the handle off to see if that helps and is not binding up with the hoses or hitting the top of the carb.
If you want to test to see if the tank is wonky just make an auxillary tank and test to see if the bike is fine without the fuel tank.
To make a super cheap auxilary McGuiver-type tank($3? maybe less):
1 get a 1 litre or 500ml water bottle. Empty it out thoroughly.
Cut a small hole in the bottom, just enough to let air in there when the bottle is upside down and feeding fuel. Tape up the hole for now with a little duct tape(handyman's secret weapon)
Go grab some cheap tubing that is thinner than the screwtop of the water bottle.
Cut a hole in the screw top of the water bottle and feed the tube through so it will let fuel come out of the tube.
Either go to a fitting shop or home depot or something to either get a brass tube that is smaller than the fuel line and the auxillary tank line or you can go fancy(the 3$) and buy a tapered fitting so you can now attach the hose from the McGuiver-type auxillary fuel tank to the fuel line using the brass fitting. Tape it up with some electrical tape or masking tape so it won't leak.
Now since this auxillary tank is going to last all of 10 minutes I go cheap and just duct tape around the hose going into the screw top so the top won't leak and it works ok(mostly). Plug the vaccuum hose with a screwdriver bit or a piece of tape.
Start the bike and hold up the auxillary tank upside down and take the tape of the little hole you made in the bottom so that air can replace the gas that is going into the carbs.
IF the bike is now perfectly fine without the tank connected then %99 it's the petcock. You may have bumped it taking it off to get to the carbs and screwed it up. Or the fuel selection lever is binding up on the crazy ass emissions hosing we are subjected to.
If you want to forgo the entire auxillary tank deal you can just start the bike with the vacuum tube plugged up and if it runs fine for the couple of minutes worth of fuel in the bowls and fuel line then it's basically the same thing your petcock is being weird. It's extremly rare that the carbs would go totally wonky just by taking them apart especially the dumping fuel problem. I'm assuming that all of the carbs are dumping fuel and not just one correct?
Also it's kind of bizarre your statement about the jets being switched. Ie one is bigger than the other and in different positions. What number carb is that? If you have the main jet where the pilot or mid jet is theoretically you would horribly overfuel that cylinder however the jets are compeletly different here is a pic of what jet is where. I would recheck that. If it's on the outside carbs you can take the bowls off with a little rachet without taking the carbs off.
Here is a pic:
As you can see the mainjet is screwed into the emulsion tube. And the brass Phillips screw above the main jet holds the floats and the needle valve in there.
Did you switch the main jet with the mid-jet on that weird carb? I hope not that will cause you problems later when the bike isn't dumping fuel. They are marked with the number of the size of the jet.
Also if you have to take the carbs off. Do yourself a huge favour and drill out the A/F mixture plugs. That way you can adjust them and if you get a jet kit in the future it will make it alot easier. Most Bandits are jetted extremely lean from the factory. Sometimes turning out the mixture screws a 1/4 turn really makes a difference in the power and drivability. Stock it's supposed to be 3 turns out, mine we all over hell's half acre from the factory and it was a little wonky driving around. Once I drilled the A/F screw plugs out and set them all the same to 3 turns out(this was before I had a jet kit) it immediately drove alot better and that was bone stock. Not only did it have a little more juice but it was alot smoother. What I did was get a drill bit that was slightly smaller than the plug I put a piece of tape on the drill bit a 1/4" from the tip so I didn't drill to deep into the screws then I drilled carefully to make the little hole in the plug a little bigger so I could screw a self taping screw in there and then got a pair of pliers and pulled the plug out. Trust me you'll thank yourself later. Remeber the plug is soft brass so it doesn't take much to drill out so be careful.