The problem is that the counterparty was (and still is) the whole host economy, the country, the world, etc. And similarly the economy would have stopped without the bailout.
There is no bigger fish when it comes to global finance, geopolitical institutions are largely reactionary (eg. the Basel III accords are nice, but they are completely post-crisis, as there was simply no incentive to push for better/stronger capital requirements before the crisis). And each country's financial regulation/oversight agency is bound by the incentives of our greedy human optimization strategies. (That is politics with all of its problems: money in politics as in regulatory capture, lack of campaign finance reforms, etc.)
https://www.nytimes.com/2008/09/26/business/26bailout.html
[“I didn’t know I was going to be the referee for an internal G.O.P. ideological civil war,” Mr. Frank said, according to The A.P.Thursday, in the Roosevelt Room after the session, the Treasury secretary, Henry M. Paulson Jr., literally bent down on one knee as he pleaded with Nancy Pelosi, the House Speaker, not to “blow it up” by withdrawing her party’s support for the package over what Ms. Pelosi derided as a Republican betrayal.]
There are some pictures of it as well but appears most of them have been taken down over the years.
Convenient
Off-topic but ultimately related, here's another "fun" Hank Paulson moment as he talks about inequality with Robert Rubin and Sheryl Sandberg (and I believe Tim Geithner is just off-camera to the left):