It doesn't sound at all like a carb problem to me; I would not touch the carb!
I would get the Kawasaki Service Manual and use the diagnostic tests in there to test the functions of the charging system and the electrical system. Replace the blown fuse. If if blows again you obviously have an electrical problem. If so, find it and fix it. In other words, instead of trying to guess at what may be causing the other problem go through and systematically correct all known problems first. Doing so may correct more than you expect.
Also, I would use the manual to perform a full tune-up on the bike. That will eliminate a bunch of variables, so if the problem persists at least you will have fewer possibilities to research. Trying to troubleshoot problems on a bike that you don't know the history of can be difficult, so doing a tune-up will help establish a new baseline from which to start. Below is a link to some info that should help you with your"new" bike. Ed
kzrider.com/forum/10-new-members/589649-...r-savannah-ga#594587