The premise of this question was, "So, suppose you’re Sleeping Beauty, and you wake up. You have no idea how the coin flip went and you have no idea what day it is. What’s the probability that the coin came up heads?"

Taking SB's perspective hypothetically--even if she's not actually in on the details, which is not clear given the above description--is a handy way to orient our thinking for analyzing the problem.

But I guess the perspective switch does introduce an additional wrinkle--or maybe pulls out the original wrinkle so we can deal with it at the time of perspective switch.

When considering the premise "suppose you're Sleeping Beauty, and you wake up" we must note that it is ambiguous whether you were SB the whole time, or your perspective just gets teleported into hers at wakeup-time. If you have a thousand rooms with a thousand SBs, and the teleporter will take you to one room, does it pick rooms at random (50% coin flip) or does it pick wakeups at random (2/3 chance for tails)?

That's basically the same statement of the non-paradox. Specify how the perspective-teleporter works, and the problem has a clear and obvious answer.