Unfortunately, for that particular Egg, you're already screwed. In Sun/Moon, Eggs are generated farther in advance than in previous generations. If the Shiny is the next Egg to come from the Daycare, it's already too late to change it without also losing it as a Shiny.
The problem with Shiny Breeding Hidden Abilities is that you aren't guaranteed to get the Hidden Ability. The best way to minimize this headache is to ensure you are using a female of the Pokemon specie with the Hidden Ability. This gives an 80% chance that the Hidden Ability will be passed down, instead of the 20% from a male.