Here are there sentences:
The position she wanted having been occupied, she failed again in her job hunting.
The position she wanted being occupied, she failed again in her job hunting.
The position she wanted occupied, she failed again in her job hunting.
In what way do the three sentences differ in meaning from one another?
I’m afraid I have not understood you the way you want me to. My concern is this: If the position is occupied, is there a meaning in saying that she has failed again.
Or, is it that the position is occupied by somebody else?
‘Occupied’ is probably not the word you require in your sentence. To indicate that someone else got the position she applied for, it would be best to use ‘offered to someone else’.
The position she wanted having been offered to someone else, she had failed again in her job hunting. (The emphasis is on the fact that the position was occupied at an earlier date than now, and remains occupied.)
The position she wanted being offered to someone else, she had failed again in her job hunting. (The emphasis is on the fact that someone else has been offered the job now.)
With the position she wanted offered to someone else, she failed again in her job hunting. (The emphasis is on the ongoing nature of the offer being given to someone other than her).
I understand that you were trying to use the different tenses, but this sentence is not straightforward and natural in any of the sentences you’ve created. They are rather false.