There are actually two good ways to shiny hunt the Regis.
Method 1: Soft-Reset
After you do the dot puzzle, save in front of the statue. Then, activate the encounter. If the Regi isn't shiny, close your game. Open it up, activate the encounter again, if it's not shiny close the game, repeat. Shininess is determined when the battle starts, not by the dots.
Method 2: Run Away
This method requires a sort of rhythm. This way, you do the dot puzzle and activate the encounter. If it's not shiny, you have to run away and do the dot puzzle again. Then repeat until you see those sparkles. If you have it down, this method can actually be faster than soft-resetting.
Unless you have the Shiny Charm, your odds of getting a shiny Regi are 1/4096.