I'm sure you know this but I'll just go over it again since this type of issue keeps coming up again and again.
You're describing a typical low speed circuit blockage and the carbs on the KZ1300 seem to be really susceptible to this. I have a feeling that it has something to do with the low speed circuit (or idle circuit as most people refer to it as) passages and how fuel hangs up in them when the bike is shut down allowing fuel to evaporate and start leaving deposits behind until the deposits clog the passages.
If you look in the throttle throat at the butterfly plates, you'll see 3 tiny ports. 1 of those ports typically sits in front of the butterfly plate when the throttle is closed and it provides fuel at throttle stop idle. The other 2 ports at throttle plate idle are hopefully not in play until you start to open up the throttle and then they add fuel to the mix. What you're describing is a blockage in the second and third port causing a lean mixture at just off idle stop until the main circuit comes in to play. Also because these are CV carbs, the main circuit is somewhat delayed until the vacuum diaphragm starts lifting the throttle barrels. So, blockage in the low speed circuit is exacerbated even more so than a standard carb.
It's my belief that cleaning out this low speed circuit is problematic due to the nature of the passages and after rebuilding carbs for 40 odd years, I learned something from Scotch. In the past, I would dunk my carbs in carb cleaner for 1-1/2 hours then blow out all the passages with compressed air, wash in hot water and blow the passages again. 95% of the time, great job and the carbs were like new again. As for the 5%, it was always a low speed circuit (off idle blockage) that wouldn't clear and no matter what I did, I couldn't clear it so carb replacement was my resolve. Now I'm talking about 5% of 400-500 carbs since I was a tuneup mechanic at a Dodge Chrysler dealer.
What was I missing that I learned from Scotch 40 years after the fact? Scotch figured out that the tight spot in the carbs was at the transition ports in the throttle bore and the only way to clear the passage was by pushing the crap backwards to the flow of fuel in the passage i.e. force the crap back out the way it came in. To do this, you need to be able to direct compress air from the throttle bore side of the transition ports, through the ports and collect the crap back in the internals of the carb. That's not an easy task without special tooling and that's where Scotch's Cleaning tool comes in. It's a brilliant design, easy to use and allows the opportunity to add Lacquer Thinner through the tool to help dissolve the crap (usually a form of shellac, varnish) and clear the passage.
Bottom line- I was a skilled carb rebuilder and through this site learned an invaluable skill that I will be forever grateful for. I too now have a "Scotch's Cleaning Tool" and plan on building a nice display rack for it and displaying in my shop front and center. For me, the Tool is a subtle reminder that you can always learn something from others and no matter how good you think you are, you can always improve so take all learning opportunities that come your way. Also when I need the tool, I know exactly where I stored it for future use