very odd problem.... I'm curious too, did you purchase a carburetor with a Kawasaki Mule part number, or one for a FE290D engine? That may be the fly in the ointment as Kawasaki made them run somewhat leaner than the general purpose version of the FE290D to conform to all 50 state versions of air quality regulations that existed then.
It also Sounds like there is a restriction, of some kind in the pilot circuit.... it's probably idled up high enough it's pulling some from the main jet as well.
Just to be sure, The jet I circled in red is the pilot jet, which you adjust until the engine reaches the highest idle, now this idle may be too high too high for it to go into gear properly so you adjust it to the 1300 rpm spec with the screw in blue