In a medium bowl, add eggs, egg yolks, sugar, flour, cornstarch and salt. Whisk until well combined. Set aside.
In a medium saucepan, combine coconut rum, cream of coconut, coconut milk and whole milk.
Place over medium heat. Bring to a boil then immediately remove from heat.
Whisking constantly, very slowly ladle about a cup of the hot rum/milk mixture into the egg mixture. Whisk quickly and well to combine.
Whisking constantly, very gradually add remaining rum/milk mixture to egg mixture in 3 or 4 additions, whisking contantly and well to combine.
Return mixture to the sauce pan. Place over medium-high heat, cook, whisking constantly, until the mixture reaches a boil and thickens, about 2 minutes. The filling must boil for at least 30 seconds in order to fully thicken.
Remove from heat and add butter. Whisk until fully incorporated.
Pour hot filling into cooled pie shell. Smooth surface with a spatula.
Press plastic wrap directly against the surface of the filling.
Place in refrigerator and cool until firm, at least 3 hours and up to 10 hours. (Any longer than 10 hours and the crust will start to soften.)