What you say sounds like fuel starvation. The biggest difference in the 400 and the 600 is that they do not have a fuel pump. Unlike the 600 the 400 has a very small inline fuel filter and a vacuum fuel tap These filters should be changed more often that the large one in the 600. It would be a good idea to change this first especially if the bike has been idle for a long time with fuel in the line. The vacuum pipe from the fuel tap goes hard and cracks or perishes and would also need to be checked. The OEM part number for the filter is 4YR-24560-00. I am sure there are inline filters available that would fit, but be sure they are good quality.
If this does not help then remove the carbs again and check the pilot jets are clear. not easiy to clean and sometime need to be soak in cellulose thinners for a few hours to dislodge the petrol laquer. If you had the carbs apart and had the main jet and the tube is screws in to, you need to check if the nozzle that fits in above the holder is still there. these do not always come easily out but can fall out
May be worthwhile sourcing a spare set of Kehin CVK CV carbs for spares, Some older small Kawasakis fours had Kehin CVK CVcarbs if you cannot get the Fazer ones
If this does not help then remove the carbs again and check the pilot jets are clear. not easiy to clean and sometime need to be soak in cellulose thinners for a few hours to dislodge the petrol laquer. If you had the carbs apart and had the main jet and the tube is screws in to, you need to check if the nozzle that fits in above the holder is still there. these do not always come easily out but can fall out
May be worthwhile sourcing a spare set of Kehin CVK CV carbs for spares, Some older small Kawasakis fours had Kehin CVK CVcarbs if you cannot get the Fazer ones