We have a set of Cayin HA-300 inhouse, so we happen to know a bit about this amplifier.
First of all, the impedance setting of HA-300 is very different from gain setting of solid state amplifier. It didn't increase or decrease the output proportionally (is that the right word to describe gain output?), it is basically the same output optimized for different impedance loading. From technical point of view, Cayin custom designed the output transformer of HA-300 with three sets of "impedance-matched" headphone output. Theoretically, specifying the nearest or matched impedance output will minimize change to headphone sound signature and deliver cleaner bass. If you are listening at regular sound level, you can switch between the impedance setting freely, it won't hurt your headphone or the amplifier, and you don't have to turn the volume down in due course.
Having said that, it's up to your personal preference to decide what works best for you. As long as you have tried the low impedance setting of HA-300 and genuinely preferred the Mid impedance setting, then you should stick with it. This is your system, so you should set it up to maximize your personal enjoyment.