In The Big Short, how did the real estate industry, banks, and investment bankers break every ethical rule, including an obligation to have personal integrity, be honest, and take responsibility for their actions? What moral philosophy did the bankers use to justify their actions and how did it lead them to behave as they did? What are two moral philosophies that would have led to a more ethical result?

In the movie, the banks and investment bankers basically collude to enrich themselves at the expense of investors.
The main culprit in this fiasco is one Michael Burry, a hedge-fund manager (played by Christian Bale). It is the year 2005, and Michael realizes that the housing market is sitting on a bubble which will eventually burst. Rather than warn the public, Michael decides to cash in on the crash when it happens. Through his research, he realizes the housing market has been supported by a glut of high-risk subprime loans. Subprime loans are typically made to investors with less-than-stellar credit, and the interest rates are usually (and sometimes substantially) higher than prime rates. These loans allow individuals with poor credit to buy houses or infuse some much-needed cash into their businesses.
These loans are extremely risky, though. When borrowers make mortgage payments on time, the loans can be extremely profitable bond assets. Because of the nature of the loan, defaults can pose an immediate threat to any investor's financial viability. Noting this, Michael decides to create what is called a default-swap market in which he can leverage his prediction of the bonds failing into a profitable enterprise. A default-swap market involves a transfer of credit risk between two parties.
The buyer is protected in case of a default or credit downgrade; he pays the seller (who assumes the credit risk) an insurance premium for this protection. For his part, the seller only needs to pay up if a definitive, negative credit event occurs. Read more about credit default swap markets here. Michael establishes a default-swap market where he bets on the probability that subprime mortgage-backed securities will fail. He is joined by Mark Baum (played by Steve Carrell) in his project.
When the housing bubble does eventually burst, Michael makes over $2.5 billion. The bursting of the housing bubble refers to a financial fallout when large numbers of individuals default on their subprime mortgages.
In the meantime, the banks and bankers Michael colludes with are guilty of ethical violations as well. They work stealthily with credit rating companies to maintain their AAA ratings until they can sell off losing positions. Michael pays the banks substantial insurance premiums in the credit-swap market, and he uses his investors' money to do this. When his investors complain and try to withdraw their money, Michael places a freeze on withdrawals. Despite this, Michael is able to profit immeasurably when the housing bubble bursts, and he manages to earn more than 500% profit for those investors who remain loyal to him throughout the financial fiasco.
On the other hand, those investors who had no inkling of the dangers of (subprime) mortgage-backed securities reaped excessive losses when the housing market destabilized. Many lost their homes and 401(K) investments. Because banks colluded with ratings companies to protect their AAA ratings on subprime loans, many investors were unaware of the risks they incurred in owning subprime-backed investments.
The moral philosophy the bankers used to justify their actions would likely be individual or moral relativism. That's the belief that all viewpoints are equally acceptable. So, an individual who is a relativist sets his own values (whatever they may be) and lives by them. In Michael Burry's case, his main aim is to make money. Although he comes to regret his actions, his initial rationalization is that wealth is a worthy goal. He profits handsomely from the financial crisis, as do his investors (he thinks this is a good thing), but he comes to realize that his willful participation in fraud has ruined many lives.
The banks and bankers who work for them justify their actions in the same way. They sell the subprime loans on the secondary market to unsuspecting investors and transfer the risks of these loans to investors. The banks rationalize that the subprime loans they originally provided helped many low-income buyers purchase their own homes.
In my opinion, two moral philosophies which would have produced more ethical results would be deontology and consequentialist utilitarianism. Deontology rests on every individual's obligation or duty to the society he/ she lives in; this moral philosophy focuses on the rightness or wrongness of one's actions. On the other hand, consequentialism pertains to the consequences of one's actions. Utilitarians believes that the most moral actions rest in bringing the greatest good to the greatest number of people.
For more, please read Deontological ethics (which also discusses deontology's foil: consequentialism).
https://www.investopedia.com/articles/investing/020115/big-short-explained.asp

https://www.philosophybasics.com/branch_deontology.html

https://www.thetruthaboutmortgage.com/what-caused-the-mortgage-crisis/