Touching on statistical matters, it is clear that he will get better with Bayern Munich than with Burnley. Bayern Munich level is still very far above Burnley, the players Bayern Munich has will make it easier for Kompany to win when facing weaker clubs. However, Kompany only challenge in his first season is when he faces another strong team or a club whose strength is almost equal to Bayern Munich, here Kompany abilities and genius will be tested. I'm not really sure that Kompany will make Bayern Munich win the Bundesliga trophy again in his first season, he needs to adapt to the new atmosphere at Bayern Munich, it takes time to achieve success, especially since the condition of the team when Kompany arrived was not in its best condition.
The thing about how Kompany will be judged is that Bayern is an unfair club for any manager out there in the world. If he wins the bundesliga, that doesn't really mean that he is a great manager, many managers could win bundesliga with Bayern, it's really not a huge achievement, sure it is good, it is better than not winning it, but it is not great for anyone, it is the expected result. Whereas, if we end up with him losing, and not winning bundesliga, he will be considered a terrible manager forever, it would take decades for him to clear his name and many successful seasons with lower teams before he can be considered for a big role again.
This is why it is not smart to accept Bayern as one of your first ever jobs, it is not going to be easy and he will not get the credit at all. Bayern is literally the biggest club in Germany, and top three or four in all of Europe, so winning something with them isn't really that impossible, it is the likely result that he will get anyway. I personally believe that we should consider how Kompany will move forward with his strategy, because if he does something unconventional for a team like Bayern and it doesn't pan out? He is going to have a very hard and difficult life.