NICK - Have you even changed the break-in oil once yet?
It takes heat and LOAD to properly break in a rebuilt Kohler. The rings have a light chrome plate on the wear surface that takes time and lots of pressure to break in properly. After the first start and carb adjustment you should run the engine at variable RPM for a half hour and re-torque the head gasket to spec. in the order the manual shows. Each bolt/nut will need some tightening. Then run the engine at full RPM at 1/4 load for an hour and increasing load after each hour to 1/2, then 3/4, then full load. During those hours you run at the suggested load for 5-10 minutes then reduce load but maintain a fast RPM. Really low or no-load operation is not recommended except for cool down before shutting the engine off. On a new tight rebuilt engine I recommend 3 to 5 minutes before shut down.
A puff of blue oil smoke on start up is not uncommon, but should clear up after a few seconds. Ring gap staggering is somewhat important but rings rotate around the piston in operation so as long as the gaps don't all align they shouldn't bother or cause smoking. There is an end gap adjustment that is critical, when up to normal operating temp if the gap is too small the ring ends could touch and that causes HUGE problems.
I rebuilt a K321 and installed it in my tractor about 4 yrs ago and it now has 150 hrs on it. I had to use it on some light spraying pulling my 55 gallon sprayer when it only had about 10-15 hrs on it and I think that light loading extended my break in period by 40-50 hours. It had about 100 hours on it before it seemed to be broke in.
If everything is correct, I wouldn't worry about a little smoke on start-up until after 50-60 hours.