1. "I have carefully taken account of all the evidence placed before me but, I am not satisfied, on the balance of probabilities, that you meet the requirements of paragraph x or y"

2. I have carefully taken into account all the evidence placed before me but, on the balance of probabilities, I am not satisfied that you meet the requirements of paragraph x or"

They are both correct. However, you need a comma before but (in both versions).

Read more about conjunctions in compound sentences.

They mean the same thing; the parenthetical phrase, 'on the balance of probabilities', is just in a different position. Whether or not it is essential, i.e. no commas are required, is questionable.