Frosted detail is made out of a acrylic polymer that has been cured with a UV lamp.
Acrylic can't be boiled or heated too much or it badly deforms. Acetone and other strong solvents also totally destroy it. Last time I used acetone on acrylic it caused it to deform and flake apart into a mush of soft acrylic chunks.
You need to use a disperse dye
A lot of instruction online involve boiling or using very hot water. I'm not sure if these instructions are fabric specific but if you boil a frosted detail printed part you'll destroy it.
Looks like Jacquard iDye Poly is the easiest one to find. It'll probably take some experimentation to find a process that works.
EDIT: see here for more details: http://textilelearner.blogspot.com/2012/01/dyeing-of-polyester-fabric-with.html