Sudah is not exactly past tense, in a grammatical sense, but it does refer to something that has happened. In this sentence, the letter was approved in the past, and now it is in a state of having been approved. So the sentence is in the present tense, but refers to something that happened in the past. It's grammatically correct and a good example of how sudah differs from "was"/"has"
ahh dude, something that has happened is called past tense. like, saya sudah hamil, saya sudah tidur.
sudah - perfective aspect, indicating that the process has been accomplished.
Sudah is often translated as already in Indonesian and Malay language textbooks to convey its meaning in the languages which express perfective aspect through inflection.
Therefore, saying Surat saya sudah disetujui is correctly translated as - My letter has already been approved.
Your own examples show that it's more complicated than that. "I am already pregnant" and "I am already asleep" are not sentences in past-tense. They refer to a present state with reference to an event that happened in the past. You probably wouldn't translate that as "I have already become pregnant"
Your translation is perfectly acceptable. In my opinion better than the one given, I just disagree with the assertion that "sudah implies past tense".