The Penguin couldn't possibly have tainted the supply of printed money by breaking into the U.S. Mint, because the Mint is only responsible for minting coins. Paper currency falls under the exclusive purview of the Bureau of Engraving and Printing.
Batman and Robin are, in Batman's own words, "duly deputized officers of the law". It is legally impossible to sue sworn law enforcement officers for false arrest - even if the arrest lacked probable cause, police officers are protected by official immunity. Had Penguin filed a lawsuit, the court would have immediately dismissed it with prejudice.
Vaccines do not offer instantaneous protection, they require weeks or longer to take effect. Even with a double dose, Penguin's gang would be no more protected than anyone else.
When at the bank Batman calls the Commissioner on a regular phone. But the Commissioner is shown speaking on the Red Direct-line Bat-Phone.
When the Penguin enters the Mint he gasses the guard at the entrance, but all four of his gang walk through the gas without any sign of being affected.