Assuming that a bit of 2x4 and a plane ticket to your outsource 'partners' is out of the question,
you may want to have a look around at something like postsharp (http://www.sharpcrafters.com/) to produce an aspect oriented solution. Basically write some code that gets to hook in to various methods. You could then wire up the events on SqlCommand; disposed, info and state change could yield something useful if you are lucky.
I would say that the best solution to to persuade your managers that a proper code fix is the only sane solution. I am very familiar with your pain but any company that allows others to write their software needs to be able to take responsibility to validate the quality of that software or be prepared to ignore the low quality and accept the consequences.
There may be mileage in identifying the really poor SQL performance code to see if that can free up some of your SQL server resource.
Good Luck