My opinion only and how I would explain it if asked:
Catcher's Obstruction is a delayed dead ball. As long as the batter tried to reach first base when she hit the ball, I believe she is OK. Since the play you presented took so long to resolve due to the overthrows, you can't offer the options to the offensive coach until the play is over. The B/R, believing she is out, heads to the dugout as usual. Since the B/R did not reach first base, you will have to offer the option to the OC.
Once the OC decided to take the penalty and you announce the award, if the B/R then decided to stay in the dugout (and I can't see why she would), you would be correct in declaring her out.
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