The modification you're proposing is analogous to a modification (I think originally due to Michael Titelbaum) called "Technicolor Beauty", in which Beauty sees either a red or a blue piece of paper in her room when she awakens on Monday (determined by a fair coin toss, independently of the "main" coin toss that decides if she's woken once or twice), and on Tuesday (if she's awakened) sees a piece of paper of whichever color she didn't see on Monday. I'll use this example rather than yours because it requires less specification about which coin toss we're talking about. Let "RB" be the hypothesis that the "main" coin toss landed Tails and Red was shown on Monday and Blue was shown on Tuesday. Let BR be the same, except Blue on Monday and Red on Tuesday.

Titelbaum used this to generate the "thirder" (SIA) answer to the problem, but SSA doesn't actually give the same answer, as you suggest it does. Even though Beauty is twice as likely to observe red paper at some point in the experiment, at no point do her conditional probabilities (e.g. for observing red, conditional on Heads or Tails) differ. Briefly: conditional on Heads, she expects to see red with probability 0.5 (because the coin toss was fair). Conditional on Tails, suppose Beauty has her eyes shut while calculating her conditional probabilities for observing red upon opening them, and evenly splits her (conditional) credence between it being Monday and it being Tuesday (SSA requires this). Now, if it's Monday, Beauty's credence in RB and BR is 0.5 for both, so she expects to see red with probability 0.5. Same goes for Tuesday.