Movies often use poetic license, departing from the facts in order to make the movie more entertaining. "The Big Short" definitely did this. I didn't see "Too Big To Fail" so I don't know how realistic their portrayal of the Lehman situation was. However, since I was net short a number of Lehman issues in 2008, I was tuned into it and here's my recollection. Google for more precise post game analysis.
Merrill Lynch and Lehman were foundering. Bank of America appeared to be interested in Lehman. I don't know if BOA was gaming the situation or not but at the last minute, they scooped up Merrill. That left Paulsen holding the bag.
Paulsen and the regulators scrambled all weekend to find another suitor. Barclays Bank was interested but wanted a lot of US financial banking. However, TARP had not yet passed so the Fed could offer none. To make things worse, it would have taken 30 days for Barclay to get shareholder approval for the deal. Unfortunately, Lehman had run out of cash and couldn't fund operations. It wasn't a question of asking them to file. With no suitors to bail them out, they HAD to declare bankruptcy.