I have never needed to put any lubricant on a case being sized in a carbide die. I think the cause of your problem is the ultrasonic cleaner.
I bought a Hornady cleaner a year or so ago and used it once. Not only did the cases not come out clean and shiny even after several cleanings, they had a sticky coating on them when they dried. I can see where that might cause problems in a die. Fortunately, I bought that ultrasonic cleaner from MidwayUSA and was able to return it for a refund.
The last thing I do to my brass is clean it and I only clean my rifle brass. After resizing, I clean or, if it is new or once-fired brass, uniform the primer pockets and trim the necks if needed. I clean the burned-on deposits from the necks with Flitz polish and then box them up until I'm ready to load them. At that time, they go into my tumbler and then go through the loading process. I load my rifle ammo on a single-stage loader.
I don't clean my handgun brass because there is too much of it and with a progressive loader, it isn't convenient unless you do it first.
Ed