Is 'I had got my hair dyed' correct?

As in British English, they use the "Have got" construction. I wonder if using the 'got' plus the 'had' for the causative is possible just like the simple present tense.

Yes, you can speak of 'getting your hair dyed' (having someone do it for you), so you can say "I had got my hair dyed" (though "I had had my hair dyed" would be more usual).