You tell the managers that you will spend no time at all on figuring out the system because you are just that good. Of course, you will still need to spend some time on it. Hopefully nobody will look into it too closely, but even if they do, you can claim that that time was already part of the "fixing the problem" phase.

Most managers will not look into the matter too closely, because if they find no wrongdoing then it was wasted time and if they do find it then they have to spend even more time to find a new freelancer. (And possibly take a reputational hit for hiring the wrong person in the first place)