I am a confirmed halfer and see no paradox here at all. Just change the "tails" rule to require 1,000,000 awakenings. When SB awakens, there's a 50% chance it's the Monday after she went to sleep, and a ~1/2,000,000 chance that it's any particular day after that. But there was always a 50% chance that the coin would be a head or a tail. In other words, I reject the idea that there are two "seemingly" valid answers. There is a right answer and an easy-to-fall-for wrong answer, full stop.

The Sleeping Beauty Problem relates to the Monty Hall problem in that in both cases, the player gets no useful information before being asked to bet on a probability. In the Sleeping Beauty problem, the player gets no information; in the Monty Hall problem, the player gets useless information, which is the same thing in logical terms.

BTW, to state the Monty Hall problem correctly, it is important to state that (i) Monty knows what's behind each door and must open a door with a goat whatever door the player picks, and (ii) that the player knows rule (i). That knowledge is analogous to SB knowing that the coin was fair, because that knowledge assures Monty's player that the information supplied by the door with the goat is useless.

SB knows all of the wake-up rules. So, too, Monty's player must know all of the door-opening rules. If, for example, Monty has discretion as to whether to open a door and offer a switch, probability analysis cannot be used to decide what to do when he does open a door with a goat.