Don't sweat it too much that you have a Dynojet kit in there. Even thought the previous owner phucked up and put a kit for another carb in there. The settings and intake apature of the Carbs are similar enough you can probably get it to work with some fiddling. The big issue is getting so that the carbs aren't being dumped a shitload of fuel into them, screwing up the jetting. Does one of the other bikes petcock work? Try the lazyman's way for now to test by just exchanging the tanks even if they are not the same colour and try the standard settings for the Dynojet kit and see what happens. That should save you a lot of time. You never know you could probably get it running pretty close to perfect you sound like you know what you are doing. The bike doesn't really care HOW it gets the fueling as long as it's correct. If you stuck a WWII German sidecar Zundapp Bing carburator on there and got it to work somehow, the bike wouldn't care as long as the fuel is metered correctly. I would imagine that once the petcock works and the standard setting on the Dynojet kit are in there, you may be slightly lean unless your at a higher altitude.
Since plugging up/clamping the the vaccuum hose and running on prime make it work better(I bet you its lean now at least when running). I would try a notch down at least on the needles and it will probably run even better. Also I would maybe try to go to the stock 1 1/2 turns out of the mixture screws as well. If you keep the tank less than half full it won't put as much pressure on those teeny float valves. Since the starting issue is better, I would guess when you stop and the fuel isn't being used it's piling up in there making the carbs so rich it will not start hot. As soon as you are running you're burning off the excess so the fueling gets better. If you exhange the tank temporarily or get a new petcock try the standard settings for the Dynojet kit which is needle at middle notch and 3 turns out or just work up from the 1 1/2 turns stock. Go for a high speed highway run as I bet your cylinders are probably black inside to clean it up, and come back home and pull the plugs and adjust as necessary.
The thing is look at it this way. My tank which is 20 litres is roughly 1kg/l or 2.2 pounds for every litre of fuel(gas is actually a little lighter) so a full tank without the petcock to regulate it is pushing down with a force of 44 pounds. I don't know of any motorcycle float valve in the world that can hold back that amount of fuel. Even only 10 litres is still 22 pounds of fuel trying to be stopped by those dinky little needle valves. It just can't handle it. How the petcock works is that vacuum is applied to a spring loaded diaphram and pulls on the diaphram depending on the engine RPM. So when you are idling it's just letting a trickle of fuel through so you aren't flooding the hell out of the carbs, at high rpm it pulls the diaphram more so that a whole bunch of fuel is allowed into the floats. With the diaphram not working and stuck open, you are basically feeding the carbs enough fuel at full throttle even though you may be at idle! Or even worse Stopped! It also prematurely wears the needle valve tips as they are being ground hard into their seats by all that fuel pressure.
No wonder it has problems starting when hot with those little dinky fuel valves are trying to hold back a full throttles worth of fuel when only sitting at idle or stopped. I would imagine if you parked the bike long enough I bet you the fuel would eventually leak into the crank case and out the airbox. The fuel valves are not designed for that pressure. However when you are running there is less pressure on the fuel valves as some of the fuel is being used to burn in the combustion chamber. Needless to say the petcock is number 1, then try to fix with what you have is number 2, and if you want to go to stock jetting ordering the jets would be number 3.
Unfortunatly from just researching this problem for you it seems that shitty petcocks is sort of characteristic of this bike. There are literally hundreds of posts out there on other forums. I guess Suzuki cut some corners on this bike to sell it cheaper. I guess that's why people spring for the fancy pingle valves. But since you will only be riding the bike for less than a year just get a wrecker petcock. If you get the bikes running well you can sell them for more money, so say a used petcock for $50 may translate into $150 bucks back when you sell it or maybe more because no one is going to buy a bike they have to fix unless it's like half off.
Put it this way:
Sell bikes as is, $1000 maybe and hard to sell, if at all most people won't touch a bike with carb/fueling issues, even if its easy to fix. Especially since these bikes are considered "beginner" bikes, chances are you are selling to a kid who wouldn't know how to fix it even if you told him how to do it. May have to sell less for parts only.
Sell bikes with $100 new petcock and jetting fix(free or $10), $2000 easy to sell, Maybe more if they're extra clean. Keep the receipts of any new parts as proof, Also a good selling feature, proves you took care of the bike. Might be worth a new petcock just to have it run awsome for a year and from an economic standpoint.
The previous owner instead of buying a Dynojet kit for $150 could have bought an aftermarket Can and slip on for $150, and $10 worth of jets and could have done it properly and gained an extra 5 HP instead of losing 5 HP. Oh well. With a little patience at the minimum you should it at least get it running as good as stock, probably better.
