I think that butterkase is usually cut to 3/8 of an inch for curd size if I remember right I don't have my notes with me.
But as far as keeping in the fat with gentle stirring I would look at it this way. The proteins have built a net around the fat globules, bacteria, and some water/whey. When you cut the curd you slice part of that net and the stuff that was trapped in the cuts is now free.
When stirring fast the curds get smooshed and out goes the fat.
So to keep it in first you need a good set, I'm not much a believer in the floc. Method, I just make a half inch cut in the curd and then stick a knife at one end and lift. If the curd breaks cleanly and is smooth and shiny it is ready.
As far as curd size you have to do some research and experiments to see what works for you.
Also something that's helps is healing the curd before stirring, this let some whey rise and gives the curds something to move around it without getting damaged.
But if you get the skin you have mentioned that traps the moisture inside you will have some problems aging the cheese and it might get acid-cut defects which is when cheeses have streaks of different colors in the paste.
That's a lot hope my rambling was somewhat helpful