As you've noticed, save states don't reset the RTC. So if Tom Nook's store opens in a half-hour and you're impatient, just hold Tab to make Dolphin run the game as fast as it can until his store opens. The RTC runs at the same speed that the game runs if you're running at 50% speed, one in-game second = two IRL seconds, and if you're running at 200% speed, one IRL second = two in-game seconds.

(This is how I never miss an in-game day in Animal Crossing, even if I manage to forget about the game for a whole IRL week.) The clock is set to your computer's clock time *only* when the game boots up – while in-game, it won't reset itself to your system time, which allows you to change the system time to whatever you want before starting the game, then change it back once the game's running without the game noticing. If you want more information, I've screwed with the real-time clock (RTC) enough in Animal Crossing (the original GC one, but this should apply to all games) that I've figured out exactly how Dolphin handles it.