I've had/still have (but I don't drive the car anymore) the same issue. From what I've found, its a big mix of a bad aux air valve, leaky injectors, bad WUR, bad fuel distributor, clogged PCV... K Jet is great and dead reliable when it's clean and in perfect working order. Otherwise you'll get weird things.
When dealing with a cold K Jet, try to start it without touching the throttle, but with the foot just above it in case it wants to stall right after starting up. Sometimes the plunger get stuck and the engine don't want to gain RPMs even with WOT, but then suddenly catches. It's really bad to deal with.
So yeah, try to start it normal the first time, having a healthy battery is important. Also bad fuel accumulator can get annoying too. If it doesn't catch on the first try, then try again but with the throttle partly depressed as stated above.
With bad K Jet, it's important to have a good ignition system. The 123 system I bought for mine helped a lot. I had really bad misfire and random stalling from time to time... Now its just running way too rich, killing plugs. Bad injectors too.
Is your idle high when warmed up ? Like 1100rpm idle when warm but barely above 600 when stone cold. It's so bad on mine that I have to sit in the car for 30s - a minute with the foot on the throttle to warm the **** up before going, or I risk stalling like a **** on the 1st few minutes of the run. I hate this feeling when the car bogs down... It should be high idle when cold and around 900 when hot. The opposite means the warming up systems aren't working and someone corrected by increasing the base idle via the screw. Not the right way to do it, but will get you there.
I'd say check the aux air valve, It's really really easy to check and It might be clogged. Also these rubber O rings around the injectors, with age they tend to leak.
If you can, get the special K Jet fuel pressure gauge. Stalling when cold can be the result of many things, the WUR is to consider... As said above, check the control pressure while the car warms up. Start it, ask for someone to held the throttle down or hold it yourself at the manifold. like 1500 rpm so it heats up. The control pressure should rise. Low control pressure = rich, high control pressure = normal. In theory. Mine is completely off. Too lean when cold and too rich when hot...
The aux air valve is heated and is opened when cold, then closes via it's heated element (or the engine's heat). They do fail/clog with time. Usually its not the only culprit.
If your K Jet is horrible when cold but fine while hot (but still drinks a lot without producing much power) then Its likely than it needs a good overall... I got tired of playing with it, let's go standalone !