The FBI does not investigate homicides within a local jurisdiction such as NYC. The NYPD would handle it.
The fact that the murders were hate crimes is the reason why the FBI was investigating this case. Had the motive not been racially motivated then it would have been investigated by the NYPD.
The agents base the arrest upon the gun that was taken from the suspect's home by his girlfriend at the direction of the FBI. Since it was his gun on his property they needed a warrant to legally obtain it. The girlfriend was acting as an agent of law enforcement which means they still needed a warrant. The case would be dismissed very quickly.