Does fast travelling affect the blood moon?
Blood moons disappear as soon as midnight hits and the cutscene activates. You must do that quest before the cutscene, while the moon itself is bright red in the sky.
After the blood moon cutscene occurs, you can watch the moon quickly fade back to its smaller, white form, which will not let you finish that shrine quest.
During the daytime, there is an NPC at Duelling Peaks Stable that will tell you if there will be a blood moon that night or not.