Nick,
In answer to your questions:
Yes, the runner must retouch home on her way back to 3rd.
As for your ruling, if she beats the throw back to 3rd, your immediate call is SAFE. In any league other than LL, I would immediately call TIME, so all the screaming about how she didn't touch home could be treated as a dead-ball appeal. Unfortunately, LL does not allow dead ball appeals, so the defense must either tag the runner or that missed base with a live ball while making the appeal.
SamC
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