[smcin]

"fixing the shrine with her gaze" totally makes sense (I'm also a native speaker of English, that makes four) and is semantically equivalent to "fixing her gaze on the shrine", it merely chooses to emphasize "the shrine" as the object, rather than "her gaze". Clearly "fix" here means "attaching securely"(/"locking on to") not "repairing", that Collins citation already given above also defined "fix" https://www.collinsdictionary.com/dictionary/english/fix

And adding the preposition "with" to get the preposition phrase "with her gaze" doesn't change the meaning: clearly the sentence is about her gaze being firmly on the shrine (not the shrine being firmly on her gaze).