"One cup of tea with milk, please."
Translation:Eine Tasse Tee mit Milch, bitte.
I deliberately left out Tasse, believing they'd require me to use "of" if I were to say Tasse Tee, and went with the hesitant presumption that saying "ein Tee" could pass for a cuppa in Germany. I was betrayed. It truly is difficult to structure phrases in my mind when I equalise things such as prepositions and conjunctions for every case; I shouldn't do this.
I've come across what you read as well. But it does not work in this case. Because, it appears as if you are asking for milk as an afterthought. This is not a standalone sentence, so to speak. Context implies that you are asking for something. As an example, consider
Gibst du bitte ihm einen Apfel? // Do you give him an apple?
Gibst du bitte mir eine Tasse Tee mit Milch? // Can you give me a glass of tea with milk?
-- These work because bitte goes with the action clause and does not break it in two.