You said that you have plenty of suction with it, so it sounds like you primed it correctly.
manufactor recommends doing a backwash prior to running the KK. I usually hook up the KK first & then backwash ( then rinse) then it works okay.

Yes you do leave the main drain on with KK ( acording to manual).
It will get stuck on the rim of your drain regardless of whether the drain is open or not ( unless your drain is actually flush with the pool bottom), you can buy a drain cover for pool cleaners that is supposed to assist it to get over the drain.
I find that if you assist it to get a lip up onto the drain, it has no problems getting over the open-running drain, so the main-drain suction should not be an issue

Hopefully by having the main-drain open, this will increase your flow enough to alieviate the air in the pump problem.
let us know if this doesn't work