A gun free bank zone would probably withstand a challenge based on Lopez. In Lopez, the Court struck down the law because the feds didn't show a nexus between the presence of firearms and an effect on interstate commerce. The claim that firearms led to violence which reduced opportunity to learn which might cause an impact on commerce which might affect interstate commerce was too tenuous.
However, banks are directly involved in interstate commerce (especially FDIC insured banks). As such, the feds could show that allowing firearms in a bank could increase the chance of a robbery, which would certainly affect interstate commerce as funds from customers from outside the state would be stolen. Further, fed funds would be required to reimburse customers. Finally, since the FBI is tasked with investigating bank robberies, there's a tacit link between the potential for crime and the possible expenditures required. This should probably get it affirmed under the Commerce Clause.
If anyone is thinking "BS", just remember that 99% of non-immigration/tax law enforcement is authorized based on the Commerce Clause. It's the most abused portion of the Constitution (even worse than the RKBA).