There are two issues here. The rating of the controller and the rating of the overloads. You have to comply with the rules for both. In general 430.83(A)(1) requires that the controller have a horsepower rating equal to or greater than the motor horsepower.
The UL standards can't over rule this requirement if this is an NEC installation. If the overload for the controller with the correct hp rating can't be set low enough to protect the motor, it sounds like a design flaw on the part of the controller manufacturer.
I agree with this opinion.
NEC tables show 15HP 460V at 21A. Ideally, if you are the manual motor controller mfr, you want the FLA of your HP rating to fall somewhere in the middle of the adjustment dial in order to allow for differences in motor designs, # of pole etc. But a mfr must also rate a device for it's switching capacity, i.e. making and breaking under load. Most likely, that particular mfr's 16 - 20A device failed the switching duty rating of a 15HP motor, otherwise they would have used it. It's a common problem for IEC manufacturers who's devices were not designed for HP rated NEMA motors, but rather IEC kW rated motors. The standard kW ratings don't line up perfectly to NEMA HP ratings.
You need to find a different manufacturer of manual motor starter, this one does not have a product that you can use.