Usually, engines are shimmed to run 15-25%, if you deviate beyond that, then reshimming may be necessary. I run 25% in everything with a medium temp plug and it works well for me.
While the SH28's I've had in the past weren't super powerful, they did hold a tune pretty well and were pretty reliable. They just didn't hold a pinch for very long. 5-6 gallons and they would start getting very fussy to tune/run/start.
I'm wondering if you have your idle set too high. A LSN will only screw in until it bottoms out in the carb slide, usually. If your idle screw has the carb too far open, then the LSN won't be able to do it's job properly.
With the throttle at neutral, and the idle screw set properly, there should be a 1mm or so gap down the throat of the carb:
The gap on this is overly wide, but it's because there's no pressure from the throttle wire pushing it shut so it's resting on the idle screw.